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RECENT DEVELOPMENTS IN CHIRAL PERTURBATION THEORY *

arXiv:hep-ph/9302247v1 11 Feb 1993RECENT DEVELOPMENTS IN CHIRAL PERTURBATION THEORY *Ulf-G. Meißner†Universit¨at Bern, Institut f¨ur Theoretische PhysikSidlerstr. 5, CH–3012 Bern, SwitzerlandAbstract: I review recent developments in chiral perturbation theory (CHPT)which is the effective field theory of the standard model below the chiral symmetrybreaking scale.

The effective chiral Lagrangian formulated in terms of the pseudoscalarGoldstone bosons (π, K, η) is briefly discussed. It is shown how one can gain insight intothe ratios of the light quark masses and to what extent these statements are model–independent.A few selected topics concerning the dynamics and interactions of theGoldstone bosons are considered.

These are ππ and πK scattering, some non–leptonickaon decays and the problem of strong pionic final state interactions. CHPT also allowsto make precise statements about the temperature dependence of QCD Green functionsand the finite size effects related to the propagation of the (almost) massless pseudoscalarmesons.

A central topic is the inclusion of matter fields, baryon CHPT. The relativisticand the heavy fermion formulation of coupling the baryons to the Goldstone fields arediscussed.

As applications, photo–nucleon processes, the πN Σ–term and non–leptonichyperon decays are presented. Implications of the spontaneously broken chiral symmetryon the nuclear forces and meson exchange currents are also described.

Finally, the use ofeffective field theory methods in the strongly coupled Higgs sector and in the calculationof oblique electroweak corrections is touched upon.Commissioned article for J. Phys. G: Nucl.

Part. Phys.BUTP–93/01January 1993* Work supported in part by Deutsche Forschungsgemeinschaft and by SchweizerischerNationalfonds.† Heisenberg Fellow.0

CONTENTS1. Introduction22.

Chiral effective Lagrangian2.1.QCD in the presence of external sources72.2.Effective theory to lowest order82.3.Chiral counting scheme102.4.Effective theory at next–to–leading order122.5.The low–energy constants162.6.Extensions193. Light quark mass ratios3.1.Lowest order estimates213.2.Next–to–leading order estimates224.

The meson sector – selected topics4.1.ππ scattering294.2.Testing the mode of quark condensation344.3.πK scattering364.4.Beyond one loop384.5.The decays K →2π and K →3π435. Finite temperatures and sizes5.1.Effective theory at finite temperature455.2.Melting condensates475.3.Effective theory in a box505.4.Finite size effects: CHPT versus Monte Carlo535.5.An application to high–Tc superconductivity546.

Baryons6.1.Relativistic formalism576.2.Non–relativistic formalism636.3.Photo–nucleon processes686.4.Baryon masses and the σ–term746.5.Non–leptonic hyperon decays776.6.Nuclear forces and exchange currents797. Strongly coupled Higgs sector7.1.Basic ideas837.2.Effective Lagrangian at next–to–leading order867.3.Longitudinal vector boson scattering887.4.Electroweak radiative corrections928.

Miscelleaneous omissions95Appendix A: The case of SU(2) × SU(2)97Appendix B: SU(3) meson–baryon lagrangian98References991

I.INTRODUCTIONEffective field theories (EFTs) have become a popular tool in particle and nuclearphysics. An effective field theory differs from a conventional renormalizable (”funda-mental”) quantum field theory in the following respect.

In EFT, one only works at lowenergies (where ”low” is defined with respect to some scale specified later) and expandsthe theory in powers of the energy/characteristic scale. In that case, renormalizabilityat all scales is not an issue and one has to handle strings of non–renormalizable interac-tions.

Therefore, at a given order in the energy expansion, the theory is specified by afinite number of coupling (low–energy) constants (this allows e.g. for an order–by–orderrenormalization).

All observables are parametrized in terms of these few constants andthus there is a host of predictions for many different processes. Obviously, at some highenergy this effective theory fails and one has to go over to a better high energy theory(which again might be an EFT of some fundamental theory).

The trace of this un-derlying high energy theory are the particular values of the low energy constants. Themost complete and most worked with EFT is chiral perturbation theory (CHPT) whichwill be the central topic of this review.

Before elaborating on the particular aspects ofCHPT, let me make some general comments concerning the applications of EFTs.Fig. 1:Scattering of light by light.

At very low photon energies, the electrons(solid lines) can be integrated out and one is left with the EFT discussed in thetext.EFTs come into play when the underlying fundamental theory contains massless (orvery light) particles. These induce poles and cuts and conventional Taylor expansions inpowers of momenta fail.

A typical example is QED where gauge invariance protects thephoton from acquiring a mass. One photon exchange involves a propagator ∼1/t, witht the invariant four–momentum transfer squared.

Such a potential can not be Taylorexpanded. Here, EFT comes into the game.

Euler and Heisenberg [1] considered the2

scattering of light by light at very low energies, ω ≪me, with ω the photon energy andme the electron mass (cf. Fig.1).

To calculate the scattering amplitude, one does notneed full QED but rather integrates out the electron from the theory. This leads to aneffective Lagrangian of the formLeff= 12( ⃗E2 −⃗B2) +e4360π2m4e( ⃗E2 −⃗B2)2 + 7( ⃗E · ⃗B)2+ .

. .

(1.1)which is nothing but a derivative expansion since ⃗E and ⃗B contain derivatives of thegauge potential. Stated differently, since the photon energy is small, the electromagneticfields are slowly varying.

From eq. (1.1) one reads offthat corrections to the leading termare suppressed by powers of (ω/me)4.

Straightforward calculation leads to the crosssection σ(ω) ∼ω6/m8e. This can, of course, also be done using full QED, but the EFTcaculation is much simpler.

A more detailed discussion of effective field theory methodsin QED can be found in the monograph by Dittrich and Reuter [2].A similar situation arises in QCD which is a non–abelian gauge theory of coloredquarks and gluons,LQCD = −14g2 GaµνGµν,a + ¯qiγµDµq −¯qMq=L0QCD+ LIQCD(1.2)with M = diag(mu, md, ms, . .

.) the quark mass matrix.For the full theory, thereis a conserved charge for every quark flavor separately since the quark masses are alldifferent.

However, for the first three flavors (u, d, s) it is legitimate to set the quarkmasses to zero since they are small on a typical hadronic scale like e.g. the ρ–mesonmass.

The absolute values of the running quark masses at 1 GeV are mu ≃5 MeV,md ≃9 MeV, ms ≃175 MeV, i.e. mu/Mρ ≃0.006, md/Mρ ≃0.012 and ms/Mρ ≃0.23[3].

If one sets the quark masses to zero, the left– and right–handed quarks defined byqL = 12(1 −γ5) q,qR = 12(1 + γ5) q(1.3)do not interact with each other and the whole theory admits an U(3) × U(3) symmetry.This is further reduced by the axial anomaly, so that the actual symmetry group ofthree flavor massless QCD isG = SU(3)L × SU(3)R × U(1)L+R(1.4)The U(1) symmetry related to baryon number conservation will not be discussed in anyfurther detail. The conserved charges which come along with the chiral SU(3) × SU(3)symmetry generate the corresponding Lie algebra.

In the sixties and seventies, manip-ulations of the commutation relations between the conserved vector (L+R) and axial–vector (L-R) charges were called ”PCAC relations” or ”current algebra calculations”3

and lead to a host of low energy theorems and predictions [4]. These rather tediousmanipulations have nowadays been replaced by EFT methods, in particular by CHPT(as will be discussed later on).

Let me come back to QCD. One quickly realizes thatthe ground state does not have the full symmetry G, eq.(1.4).

If that were the case,every known hadron would have a partner of the same mass but with opposite parity.Clearly, this is in contradiction with the observed particle spectrum. Another argumentinvolves the two–point correlation functions of the vector (V aµ ) and axial–vector (Aaµ)currents.

These can be determined from the semi–leptonic τ decays τ →nπ + ντ [5]and they show a rather different behaviour. While the VV–correlator is peaked aroundMρ ≃770 MeV and smooth otherwise, the AA–correlator shows a broad enhancementaround MA1 ≃1200 MeV and is smooth elsewhere.

The physical ground state musttherefore be asymmetric under the chiral SU(3)L × SU(3)R [6]. In fact, the chiral sym-metry is spontaneously broken down (hidden) to the vectorial subgroup of isospin andhypercharge, generated by the vector currents,H = SU(3)L+R × U(1)L+R(1.5)As mandated by Goldstone’s theorem [7], the spectrum of massless QCD must there-fore contain N 2f −1 = 9 −1 = 8 massless bosons with quantum numbers JP = 0−(pseudoscalars) since the axial charges do not annihilate the vacuum.

Reality is a bitmore complex. The quark masses are not exactly zero which gives rise to an explicitchiral symmetry breaking (as indicated by the term LIQCD in eq.(1.2)).

This is in agree-ment with the observed particle spectrum – there are no massless strongly interactingparticles. However, the eight lightest hadrons are indeed pseudoscalar mesons.

Theseare the pions (π± , π0), the kaons (K± , ¯K0 , K0) and the eta (η). One observes thatMπ ≪MK ≈Mη which indicates that the masses of the quarks in the SU(2) subgroup(of isospin) should be considerably smaller than the strange quark mass.

This expecta-tion is borne out by actual calculation of quark mass ratios as discussed later on. Also,from the relative size of the quark masses mu,d ≪ms one expects the chiral expansionto converge much more rapidly in the two–flavor case than for SU(3)f. These basicfeatures of QCD can now be explored in a similar fashion as outlined before for the caseof QED.

The technical details will be given in later sections, here I just wish to outlinethe main ideas beyond CHPT.As already noted, the use of EFTs in the context of strong interactions preceedsQCD. The Ward identities related to the spontaneously broken chiral symmetry wereexplored in great detail in the sixties in the context of current algebra and pion poledominance [4,8].

The work of Dashen and Weinstein [9], Weinberg [10] and Callan,Coleman, Wess and Zumino [11] clarified the relation between current algebra calcula-tions and the use of effective Lagrangians (at tree level). However, only with Weinberg’s[12] seminal paper in 1979 it became clear how one could systematically generate loopcorrections to the tree level (current algebra) results.

In fact, he showed that theseloop corrections are suppressed by powers of (E/Λ)2, with E a typical energy (four–momentum) and Λ the scale below which the EFT can be applied (typically the mass of4

the first non–Goldstone resonance, in QCD Λ ≃Mρ). The method was systematized byGasser and Leutwyler for SU(2)f in Ref.

[13] and for SU(3)f in Ref. [14] and has becomeincreasingly popular ever since.

The basic idea of using an effective Lagrangian insteadof the full theory is based on a universality theorem for low energy properties of fieldtheories containing massless (or very light) particles. Consider a theory (like QCD) atlow energies.

It exhibits the following properties:• L is symmetric under some Lie group G (in QCD: G = SU(3)L × SU(3)R).• The ground state |0 > is symmetric under H ⊂G (in QCD: H = SU(3)V ). To anybroken generator of G there appears a massless Goldstone boson (called ”pion”)with the corresponding quantum numbers (JP = 0−in QCD).• The Goldstone bosons have a finite transition amplitude to decay into the vacuum(via the current associated with the broken generators).

This matrix element carriesa scale F, which is of fundamental importance for the low energy sector of thetheory (in QCD: < 0|Aaµ|πb >= ipµδabF, with F the pion decay constant in thechiral limit).• There exists no other massless (strongly interacting) particles.• Explicit symmetry breaking (like the quark mass term in QCD) can be treated ina perturbative fashion.Now any theory with these properties looks the same (in more than two space-time dimensions). This means that to leading order the solution to the Ward identitiesconnected to the broken symmetry is unique and only contains the scale F.Thus,the EFT to lowest order is uniquely fixed and it is most economical to formulate it interms of the Goldstone fields.

In fact, one collects the pions in a matrix–valued function(generally denoted ’U’) which transforms linearly under the full action of G. In QCD,a popular choice is U(x) = exp[iλaπa(x)/F] with λa ( a = 1, . .

., 8) the Gell–Mannmatrices and U ′(x) = RU(x)L† under chiral SU(3)L ×SU(3)R (with L, R an element ofSU(3)L,R). Accordingly, the pion fields transform in a highly non–linear fashion [10,11].This is a characteristic feature of EFTs.While the discussion so far was mostly centered around QCD, the universality the-orem immediately allows one to relate QCD to the Higgs model of standard electroweaksymmetry breaking.

The Higgs model has G = O(4) broken down to H = O(3), which isa structure isomorphic to two–flavour massless QCD. The only difference is that whilein QCD one has F ≃93 MeV, in the Higgs model F ≃250 GeV.

The longitudinaldegrees of freedom of the W ± and Z0 bosons play the role of the pions as discussedin more detail later on. Where these two theories really differ is at next–to–leadingorder.

While to leading order all Green functions are given in terms of the scale F (andby some parameters related to explicit symmetry breaking), the solutions to the Wardidentities at next–to–leading order can only be given modulo some unknown coefficients,the low–energy constants. These have two purposes: First, they allow to absorb the di-vergences inflicted by the loop contributions which have to be accounted for and, second,5

their finite values reflect the underlying high energy theory. In QCD, these low energyconstants are functions of the scale ΛQCD and the masses of the heavy quarks and arein principle calculable.

However, at present one has to determine them phenomeno-logically or use some models to estimate them. This will be discussed in the followingsection.

It is important to notice that in this way one can set up a consistent order-by-order renormalization scheme. This allows one to make predictions once one has useda certain set of processes to pin down the low–energy constants.

In what follows, I willalways consider EFTs beyond tree level, i.e. taking into account loop corrections.

Animportant argument for doing this is unitarity. Tree graphs are always real and thusunitarity is violated.

Evidently, loop graphs have imaginary parts and one can restoreunitarity in a perturbative fashion. It was already noted long ago that there are largeunitarity corrections to some processes [15], but it should be stressed that there are alsoother corrections.

CHPT accounts for all of them in a systematic manner. The generalframework of EFTs is therefore based on some general principles like Lorentz invariance,analyticity, unitarity and cluster decomposition.

If one has a finite number of masslessor light particle types, one simply writes down an effective non–linear Lagrangian ofthese particles in accordance with the pertinent symmetry requirements. Correctionsto the leading order predictions can be worked out in a systematic fashion.

For doingthat, one invokes higher dimensional operators whose contributions are suppressed bypowers of 1/Mnew, with Mnew a typical scale of new physics (like the ρ in QCD or thetechni-rho in technicolor). As we will see in what follows, there exists special cases werethe one loop corrections are already large at low energies.

This forces one to go beyondthe generally well working one loop approximation (as will be discussed in some detaillater on).We are now in the position to consider more specific examples build on these generalideas. The material is organized as follows.

In section II, I review the basic constructionof the next–to–leading order chiral Lagrangian of the strong interactions. It also containsthe pertinent chiral counting rules which are at the heart of the systematic expansionin small momenta and quark masses.

Extensions to incorporate the energy–momentumtensor and the U(1)A anomaly are briefly touched upon. The physics related to theanomalous sector (Wess-Zumino-Witten term) is not discussed in any detail.

In sectionIII, it is shown how chiral perturbation theory can be used to gain insight about theratios of the light quark masses.Section IV is concerned with some applications inthe meson sector like the classical field of pion–pion scattering. I also elucidate someproblems due to strong pionic final state interactions.

The extension to non-leptonicweak decays is briefly discussed in the context of the decays K →2π, 3π. In sectionV effects of finite temperatures and volumes are considered.

In particular, the meltingof the quark and gluon condensates and finite size effects in relation to lattice gaugetheories are discussed. The following section VI is devoted to the field of baryon CHPT,with special emphasis on the newly developed methods from heavy quark EFTs andapplications to photo-nucleon processes.

In section VII, it is shown how CHPT methodscan be used in the strongly interacting Higgs sector. The topics of WLWL scattering and6

oblique corrections to the Z propagator are discussed. In section VIII, miscellaneousapplications and developments are touched upon.II.CHIRAL EFFECTIVE LAGRANGIANIn this section, I briefly review how to construct the effective chiral Lagrangian ofthe strong interactions at next–to–leading order, following closely the work of Gasserand Leutwyler [14].It is most economical to use the external field technique sinceit avoids any complication related to the non–linear transformation properties of thepions.The basic objects to consider are currents and densities with external fieldscoupled to them [16] in accordance with the symmetry requirements.

The associatedGreen functions automatically obey the pertinent Ward identities and higher derivativeterms can be constructed systematically. The S–matrix elements for processes involvingphysical mesons follow then via standard LSZ reduction.2.1.

QCD in the presence of external sourcesConsider the vacuum–to–vacuum transition amplitude in the presence of externalfieldseiZ[v,a,s,p] =< 0 out|0 in >v,a,s,p(2.1)based on the QCD LagrangianL = L0QCD + ¯q(γµvµ(x) + γ5γµaµ(x))q −¯q(s(x) −ip(x))q −θ(x)16π2 Gaµν ˜Gµν,aL0QCD = −12g2 GaµνGµν,a + ¯qiγµ(∂µ −iAµ)q(2.2)with Aµ the gluon field, Gµν the corresponding field strength tensor and ˜Gµν =12ǫµναβGαβ its dual. L0QCD is massless QCD with vanishing vacuum angle.

The ex-ternal vector (vµ), axial–vector(aµ), pseudoscalar (p) and scalar (s) fields are hermitean3 × 3 matrices in flavor space. The quark mass matrix M,M = diag(mu , md , ms)(2.3)is contained in the scalar field s(x).

The Green functions of massless QCD are obtainedby expanding the generating functional around vµ = aµ = s = p = 0 and θ(x) = θ0. Forthe real world, one has to expand around vµ = aµ = p = 0 , s(x) = M and θ(x) = θ0.The Lagrangian L is invariant under local SU(3) × SU(3) chiral transformations if thequark and external fields transform as follows:q′R = Rq ;q′L = Lqv′µ + a′µ = R(vµ + aµ)R† + iR∂µR†v′µ −a′µ = L(vµ −aµ)L† + iL∂µL†s′ + ip′ = R(s + ip)L†(2.4)7

with L, R elements of SU(3)L,R (in general, these are elements of U(3)L,R, but wealready account for the axial anomaly to be discussed later). The path integral repre-sentation of Z reads:eiZ[v,a,s,p] =Z[DAµ][Dq][D¯q]eRid4xL(q,¯q,Gµν ; v,a,s,p)(2.5)It allows one to make contact to the effective meson theory.

Since we are interestedin processes were the momenta are small (the low energy sector of the theory), we canexpand the Green functions in powers of the external momenta. This amounts to anexpansion in derivatives of the external fields.

As already pointed out, the low energyexpansion is not a simple Taylor expansion since the Goldstone bosons generate polesat q2 = 0 (in the chiral limit) or q2 = M 2π (for finite quark masses). The low energyexpansion involves two small parameters, the external momenta q and the quark massesM.

One expands in powers of these with the ratio M/q2 fixed. The effective mesonLagrangian to carry out this procedure follows from the low energy representation ofthe generating functionaleiZ[v,a,s,p] =Z[DU]eRid4xLeff(U ; v,a,s,p)(2.6)where the matrix U collects the meson fields.

The low energy expansion is now obtainedfrom a perturbative expansion of the meson EFT,Leff= L2 + L4 + . .

. (2.7)where the subscript (n = 2, 4, .

. .) denotes the low energy dimension (number of deriva-tives and/or quark mass terms).

The various terms in this expansion will now be dis-cussed.2.2. Effective theory to leading orderLet us now construct the leading term (called L2) in the low energy expansion (2.7).The mesons are described by a unitary 3 × 3 matrix in flavor space,U †U = 1 ,det U = 1(2.8)The matrix U transforms linearly under chiral symmetry, U ′ = RUL†.

The lowest orderLagrangian consistent with Lorentz invariance, chiral symmetry, parity, G–parity andcharge conjugation reads [14]L = 14F 2Tr[∇µU †∇µU + χ†U + χU †]+ 112H0∇µθ∇µθ(2.9)8

The covariant derivative ∇µU transforms linearly under chiral SU(3) × SU(3) and con-tains the couplings to the external vector and axial fields,∇µU = ∂µU −i(vµ + aµ)U + iU(vµ −aµ)(2.10)The field χ embodies the scalar and pseudoscalar externals,χ = 2B(s + ip)(2.11)The last term in (2.9) is related to CP violation in the strong interaction.For themoment, we will set this term to zero (which agrees with the empirical observationthat the electric dipole moment of the neutron is tiny). Let us now discuss the variousconstants appearing in eqs.(2.9,2.11).

The scale F is related to the axial vector currents,Aaµ = −F∂µπa + . .

. (2.12)and thus can be identified with the pion decay constant in the chiral limit, F = Fπ{1 +O(M)}, by direct comparison with the matrix–element < 0|Aaµ|πb >= ipµδabF.

Theconstant B, which appears in the field χ, is related to the explicit chiral symmetrybreaking. Consider the symmetry breaking part of the Lagrangian and expand it inpowers of the pion fields (with p = 0, s = M so that χ = 2BM)*LSB2= 12F 2BTr[M(U + U †)] = (mu + md + ms)B[F 2 −π22 +π424F 2 + .

. .

](2.13)with π = λaπa. The first term on the right hand side of eq.

(2.13) is obviously related tothe vacuum energy, while the second and third are meson mass and interaction terms,respectively. Since ∂HQCD/∂mq = ¯qq it follows from (2.13) that< 0|¯uu|0 >=< 0| ¯dd|0 >=< 0|¯ss|0 >= −F 2B{1 + O(M)}(2.14)This shows that the constant B is related to the vev’s of the scalar quark densities< 0|¯qq|0 >, the order parameter of the spontaneous chiral symmetry breaking.

Therelation (2.14) is only correct modulo higher order corrections in the quark masses asindicated by the term O(M). One can furthermore read offthe pseudoscalar mass termsfrom (2.13).

In the case of isospin symmetry (mu = md = ˆm), one findsM 2π = 2 ˆmB{1 + O(M)} =◦M2π{1 + O(M)}M 2K = ( ˆm + ms)B{1 + O(M)} =◦M2K{1 + O(M)}M 2η = 23( ˆm + 2ms)B{1 + O(M)} =◦M2η{1 + O(M)}(2.15)* remember that π stands as a generic symbol for the pions, kaons and the η.9

with◦M P denoting the leading term in the quark mass expansion of the pseudoscalarmeson masses. For the◦M P , the Gell–Mann–Okubo relation is exact, 4◦M2K =◦M2π+3◦M2η[17].

In the case of isospin breaking, which leads to π0 −η mixing, these mass formulaeare somewhat more complicated (see section 3 and ref.[14]). Eq.

(1.15) exhibits nicelythe Goldstone character of the pions – when the quark masses are set to zero, thepseudoscalars are massless and SU(3)×SU(3) is an exact symmetry. For small symmetrybreaking, the mass of the pions is proportional to the square root of the symmetrybreaking parameter, i.e.

the quark masses (an alternative scenario is discussed in section4.2). From eqs.

(2.13) and (2.15) one can eliminate the constant B and gets the celebratedGell–Mann–Oakes–Renner [18] relationsF 2πM 2π = −2 ˆm < 0|¯uu|0 > +O(M2)F 2KM 2K = −( ˆm + ms) < 0|¯uu|0 > +O(M2)F 2η M 2η = −23( ˆm + ms) < 0|¯uu|0 > +O(M2)(2.16)where we have used FP = F{1 + O(M)} (P = π, K, η), i.e. the differences in thephysical decay constants Fπ ̸= FK ̸= Fη appear in the terms of order M2.From this discussion we realize that to leading order the strong interactions arecharacterized by two scales, namely F and B. Numerically, using the sum rule value< 0|¯uu|0 >= (−225 MeV)3 [19], one hasF ≃Fπ ≃93 MeVB ≃1300 MeV(2.17)The large value of the ratio B/F ≃14 has triggered some investigations of alternativescenarios concerning the mode of quark condensation as will be discussed later.One can now calculate tree diagrams using the effective Lagrangian L2 and derivewith ease all so–called current algebra predictions (low energy theorems).

Current al-gebra is, as should have become evident by now, only the first term in a systematiclow energy expansion. Working out tree graphs using L2 can not be the whole story –tree diagrams are always real and thus unitarity is violated.

One has to include higherorder corrections to cure this. To do this in a consistent fashion, one needs a countingscheme to be discussed next.2.3.

Chiral counting schemeThe leading term in the low energy expansion of Leff(2.7) was denoted L2 because ithas dimension (chiral power) two. It contains two derivatives or one power of the quarkmass matrix.

If one assumes the matrix U to be order one, U = O(1), a consistentpower counting scheme for local terms containing U, ∂µU, vµ, aµ, s, p, . .

. goes asfollows.Denote by q a generic small momentum (for an exact definition of ’small’,10

see eq.(2.27)). Derivatives count as order q and so do the external fields which occurlinearly in the covariant derivative ∇µU.

For the scalar and pseudoscalar fields, it ismost convenient to book them as order q2. This can be traced back to the fact thatthe scalar field s(x) contains the quark mass matrix, thus s(x) ∼M ∼M 2π ∼q2.

Withthese rules, all terms appearing in (2.9) are of order q2, thus the notation L2 (noticethat a term of order one is a constant since U †U = 1 and can therefore be disregarded.Odd powers of q clash with parity requirements). To summarize, the building blocks ofall terms containing derivatives and/or quark masses have the following dimension:∂µU(x) , vµ(x) , aµ(x) = O(q)s(x) , p(x) , F L,Rµν (x) = O(q2)(2.18)where we have introduced the field strength F L,Rµνfor later use.

They are defined viaF Iµν = ∂µF Iν −∂νF Iµ −i[F Iµ, F Iν ] , I = L, RF Rµν = vµ + aµ ;F Lµν = vµ −aµ(2.19)As already mentioned, unitarity calls for pion loop graphs. Weinberg [12] made theimportant observation that diagrams with n (n = 1, 2, .

. .) meson loops are suppressedby powers of (q2)n with respect to the leading term.

His rather elegant argument goesas follows. Consider the S–matrix for a reaction involving Ne external pionsS = δ(p1 + p2 + .

. .

+ pNe)M(2.20)with M the transition amplitude.The dimension of M is [M] = 4 −Ne since anyexternal pion wave function scales like [mass]−1 according to the usual PCAC relation.Now M depends on the total momentum flowing through the amplitude, on the pertinentcoupling constants g and the renormalization scale µ (the loop diagrams are in generaldivergent and need to be regularized),M = M(q , g , µ) = qDf(q/µ , g)D = 4 −Ne −∆(2.21)Here, ∆is the dimension of the different couplings and propagators entering the tran-sition amplitude.For the effective meson theories under consideration, the couplingconstants associated with the pionic interactions scale as 4 −d, with d the number ofderivatives as mandated by Goldstone’s theorem (in the chiral limit). If M contains Ndsuch vertices and Ni internal propagators, we have∆=XdNd(d −4) −2Ni −Ne(2.22)11

The total number of loops (Nl) follows from simple topological arguments to beNe = Ni −XdNd −1(2.23)so that the total scaling dimension D of M isD = 2 +XdNd(d −2) + 2Nl(2.24)The dominant graphs at low energy carry the smallest value of D. The leading termswith d = 2 scale like q2 at tree level (Nl = 0), like q4 at one loop level (Nl = 1) andso on. Higher derivative terms with d = 4 scale as q4 at tree level, as q6 at one–looporder etc.

This power suppression of loop diagrams is at the heart of the low energyexpansion in EFTs like e.g. CHPT.Up to now, I have been rather casual with the meaning of the word ”small”.

Bysmall momentum or small quark mass I mean this with respect to some typical hadronicscale, also called the scale of chiral symmetry breaking (denoted by Λχ). Georgi andManohar [20] have argued that a consistent chiral expansion is possible ifΛχ ≤4πFπ ≃1 GeV(2.25)Their argument is based on the observation that under a change of the renormalizationscale of order one typical loop contributions (say to the ππ scattering amplitude) willcorrespond to changes in the effective couplings of the order F 2π/Λ2χ ≃1/(4π)2.

Set-ting Λχ = 4πFπ and cutting the logarithmically divergent loop integrals at this scale,quantum corrections are of the same order of magnitude as changes in the renormalizedinteraction terms. The factor (4π)2 is related to the dimensionality of the loop integrals(for a more detailed outline of this argument see the monograph by Georgi [21]).

An-other type of argument was already mentioned in section 1. Consider ππ scattering inthe I = J = 1 channel.

There, at √s = 770 MeV, one hits the ρ–resonance. This is anatural barrier to the derivative expansion of the Goldstone mesons and therefore servesas a cut off.

The appearance of the ρ signals the regime of the non–Goldstone particlesand describes new physics. It is therefore appropriate to chooseΛχ ≃Mρ ≃770 MeV(2.26)which is not terribly different from the estimate (2.25).

In summary, small externalmomenta q and small quark masses M meansq/Mρ ≪1 ;M/Mρ ≪1 . (2.27)2.4.

Effective theory at next–to–leading order12

We have now assembled all tools to discuss the generating functional Z at next–to–leading order, i.e. at O(q4).

It consists of three different contributions:1) The anomaly functional is of order q4 (it contains four derivatives). We denotethe corresponding functional by ZA.

The explicit construction was given by Wessand Zumino [22] and can also be found in ref.[14]. A geometric interpretation isprovided by Witten [23].2) The most general effective Lagrangian of order q4 which is gauge invariant.

It leadsto the action Z2 + Z4 =Rd4xL2 +Rd4xL4.3) One loop graphs associated with the lowest order term, L2. These also scale asterms of order q4.Let me first discuss the anomaly functional ZA.

It subsumes all interactions whichbreak the intrinsic parity and is responsible e.g. for the decay π0 →2γ.

It also generatesinteractions between five or more Goldstone bosons [23]. In what follows, we will notconsider this sector in any detail (for a review, see ref.

[24] and recent work in the contextof CHPT is quoted in section 8).What is now the most general Lagrangian at order q4? The building blocks and theirlow energy dimensions were already discussed – we can have terms with four derivativesand one quark mass or with two quark masses (and, correspondingly, the other externalfields).In SU(3), the only invariant tensors are gµν and ǫµναβ, so one is left with(imposing also P, G and gauge invariance)L =10Xi=1LiPi +2Xj=1Hj ˜Pj(2.28)withP1 = Tr (∇µU †∇µU)2P2 = Tr (∇µU †∇νU) Tr (∇µU †∇νU)P3 = Tr (∇µU †∇µU∇νU †∇νU)P4 = Tr (∇µU †∇µU) Tr (χ†U + χU †)P5 = Tr (∇µU †∇µU)(χ†U + χU †)P6 =Tr (χ†U + χU †)2P7 =Tr (χ†U −χU †)2P8 = Tr (χ†Uχ†U + χU †χU †)P9 = −i Tr (F Rµν∇µU∇νU †) Tr (F Lµν∇µU †∇νU)P10 = Tr (U †F RµνUF L,µν)˜P1 = Tr (F RµνUF R,µν + F LµνUF L,µν)˜P2 = Tr (χ†χ)(2.29)13

For the two flavor case, not all of these terms are independent. The pertinent q4 effectiveLagrangian is discussed in Appendix A.

The first ten terms of (2.28) are of physicalrelevance for the low energy sector, the last two are only necessary for the consistentrenormalization procedure discussed below. These terms proportional to ˜Pj(j = 1, 2)do not contain the Goldstone fields and are therefore not directly measurable at lowenergies.

The constants Li (i = 1, . .

., 10) appearing in (2.28) are the so–called low–energy constants. They are not fixed by the symmetry and have the generic structureLi = Lri + Linfi(2.30)These constants serve to renormalize the infinities of the pion loops (Linfi ) and theremaining finite pieces (Lri ) have to be fixed phenomenologically or to be estimated bysome model (see below).

At next–to–leading order, the strong interactions dynamics istherefore determined in terms of twelve parameters – B, F, L1, . .

., L10 (remember thatwe have disregarded the singlet vector and axial currents). In the absence of externalfields, only the first three terms in (2.28) have to be retained.Finally, we have to consider the loops generated by the lowest order effective La-grangian.

These are of dimension q4 (one loop approximation) as mandated by Wein-berg’s scaling rule.To evaluate these loop graphs one considers the neighbourhoodof the solution ¯U(x) to the classical equations of motion. In terms of the generatingfunctional, this readseiZ = eiRd4x[L2( ¯U)+L4( ¯U)]Z[DU]eiRd4x[L2(U)−L2( ¯U)](2.31)The bar indicates that the Lagrangian is evaluated at the classical solution.

Accordingto the chiral counting, in the second factor of (2.31) only the term L2 is kept. This leadstoZ =Zd4x( ¯L2 + ¯L4) + i2ln detD(2.32)The operator D is singular at short distances.

The ultraviolet divergences containedin ln detD can be determined via the heat kernel expansion [25]. Using dimensionalregularization, the UV divergences in four dimensions take the form−1(4π)21d −4Sp(12 ˆσ2 + 112ˆΓµν ˆΓµν)(2.33)The explicit form of the operators ˆσ and ˆΓµν can be found in ref.[14].

Using their explicitexpressions, the poles in ln detD can be absorbed by the following renormalization ofthe low energy constants:Li = Lri + Γiλ,i = 1, . .

., 10Hj = Hrj + ˜Γjλ,j = 1, 2(2.34)14

withλ =116π2 µd−41d −4 −12[ln(4π) + Γ′(1) + 1]Γ1 = 332 ,Γ2 = 316 ,Γ3 = 0 ,Γ4 = 18 ,Γ5 = 38 ,Γ6 = 11144 ,Γ7 = 0 ,Γ8 = 548 ,Γ9 = 14 ,Γ10 = −14 ,˜Γ1 = −18 ,˜Γ2 = 524 ,(2.35)and µ is the scale of dimensional regularization. The q4 contribution Z4 + Z1−loop isfinite at d = 4 when expressed in terms of the renormalized coupling constants Lri andHri .

The next step consists in the expansion of the differential operator D in powersof the external fields. This gives the explicit contributions of the one–loop graphs to agiven Green function.

The full machinery is spelled out in Gasser and Leutwyler [14].In general, one groups the loop contributions into tadpole and unitarity corrections.While the tadpoles contain one vertex and one loop, the unitarity corrections containone loop and two vertices.The tadpole contributions renormalize the couplings ofthe effective Lagrangian. Both of these loop contributions also depend on the scale ofdimensional regularization.

In contrast, physical observables are µ–independent. Foractual calculations, however, it is sometimes convenient to choose a particular value ofµ, say, µ = Mη or µ = Mρ.Fig.

2:Goldstone boson scattering in the one–loop approximation. At lowestorder only tree diagrams (a) contribute.At next–to–leading order, one hastadpole (b), unitarity (c) and higher derivative (as denoted by the black box in(d)) corrections.15

To one–loop order, the generating functional therefore takes the formZ = Z2 + Z4 + Zone−loop + Zanom(2.36)and what remains to be done is to determine the values of the renormalized low energyconstants, Lri (i = 1, . .

., 10). Before doing this, let me graphically show the one–loopapproximation to the ππ–scattering amplitude (fig.2).

The first term on the right handside is the famous tree level term which leads to Weinberg’s current algebra predictionfor the scattering amplitude.The next two terms depict a tadpole and a unitaritycorrection. Finally, the term with the black box is a higher derivative term accompaniedby a low energy constant.

The analytical form of this amplitude will be discussed insection 4.2.5. The low–energy constantsThe low energy constants are in principle calculable from QCD, they depend onΛQCD and the heavy quark massesLri = Lri (ΛQCD ; mc , mb , mt)(2.37)In practice, such a calculation is not feasible.

One therefore resorts to phenomenologyand determines the Lri from data.However, some of these constants are not easilyextracted from empirical information. Therefore, one uses constraints from the large Ncworld [16,27].

In the limit of Nc (with Nc the number of colors) going to infinity andkeeping g2Nc fixed, the Green functions are proportional to some power of Nc [26, 27,28]. Furthermore, in this limit the η′ becomes massless, Mη′ ∼1/Nc.

This leads to anenhancement of the coupling L7,Lη′7 = −γ248F 2M 2η(2.38)with γ measuring the strength of ηη′ mixing. The large–Nc counting rules for all Lrihave been worked out by Gasser and Leutwyler [14]:O(N 2c ) :L7O(Nc) :L1 , L2 , L3 , L4 , L8 , L10O(1) :2L1 −L2 , L4 , L6(2.39)Using this and experimental information from ππ scattering, FK/Fπ, the electromag-netic radius of the pion and so on, one ends up with the values for the Lri (µ = Mη) givenin table 1 (large Nc arguments are used to estimate L1 , L4 and L6 ).

For comparison,we also give the values at µ = Mρ. It should be noted that the large Nc suppressionof 2L1 −L2 has recently been tested using data from Kℓ4 decay.

Riggenbach et al.16

[29] find (2L1 −L2)/L3 = −0.19+0.55−0.80, i.e. the large Nc prediction works within onestandard deviation.

More accurate data will allow to further pin down this quantity. Inthe case of SU(2), one can define scale–independent couplings ¯ℓi (i = 1, .

. ., 7).

Theseare discussed in appendix A.Can one now understand the values of the Lri from some first principles? Alreadyin their 1984 paper, Gasser and Leutwyler [13] made the following observation.

Theyconsidered an effective theory of ρ mesons coupled to the pseudoscalars. Eliminatingthe heavy field by use of the equations of motion in the region of momenta much smallerthan the ρ mass, one ends up with terms of order q4.

The values of the correspondinglow energy constants are given in terms of Mρ and the ρ–meson coupling strengths tophotons and pions. This leads to a fair description of the low energy constants.

Tomake this statement more transparent, consider the vector form factor of the pion< π+(p′)|Jemµ |π+(p) >= FV (t)(p + p′)µ(2.40)with t = (p′ −p)2. For small t, the form factor can be Taylor expanded,FV (t) = 1 + 16 < r2 >πV t + O(t2)(2.41)Calculating FV (t) with L2 alone, one gets FV (t) = 1.

The one–loop result has the formFV (t) = 1 +2L9F 2π−196π2F 2πlnM 2πµ2−13t+ t −4M 2π96π2F 2π2 + σ lnσ −1σ + 1 + iπΘ(t −4M 2π)(2.42)with σ =p1 −4M 2π/t. The last term contains the imaginary part required by unitarityand some rescattering corrections.

Clearly, one has to pin down Lr9 to have a predictionfor the form factor. As it is well known, FV (t) is well described within the vector mesondominance (VMD) picture,FV (t) =M 2ρM 2ρ −t = 1 +tM 2ρ+ t2M 4ρ+ .

. .

(2.43)where I have neglected imaginary parts and alike (a better form is given in Ref. [30]).Of course, the expansion in t/M 2ρ is only useful as long as t/M 2ρ ≪1.

Comparing theterms linear in t, we find from (2.42) and (2.43) that Lr9 = F 2π/2M 2ρ ≃7.3 · 10−3, whichis close to the phenomenological value of Lr9(Mρ). The small discrepancy is due to thefact that VMD does not accurately give the vector radius, < r2 >Vπ , VMD= 6/M 2ρ = 0.40fm2 while experimentally < r2 >Vπ = 0.439 ± 0.008 fm2 [31].17

This method has been generalized by Ecker et al. [32] and by Donoghue et al.[33].

They consider the lowest order effective theory of Goldstone bosons coupled toresonance fields (R). These resonances are of vector (V), axial–vector (A), scalar (S)and non–Goldstone pseudoscalar (P) type.

For the latter category, only the η′ is ofpractical importance (cf. eq(2.38)).

The form of the pertinent couplings is dictated bychiral symmetry in terms of a few coupling constants which can be determined fromdata (from meson–meson and meson–photon decays). At low momenta, one integratesout the resonance fields.

Since their couplings to the Goldstone bosons are of order q2,resonance exchange produces terms of order q4 and higher. Symbolically, this readsZ[dR]expiZd4x ˜Leff[U, R]= expiZd4xLeff[U](2.44)So to leading order (q4), one only sees the momentum–independent part of the resonancepropagators,1M 2R −t =1M 2R1 +tM 2R −t(2.45)and thus the Lri (µ ≃MR) can be expanded in terms of the resonance coupling constantsand their masses.

This leads toLri (µ) =XR=V,A,S,PLResi+ ˆLi(µ)(2.46)with ˆLi(µ) a remainder. For this scenario to make sense, one has to choose µ somewherein the resonance region.

A preferred choice is µ = Mρ (as shown in Ref. [32], any valueof µ between 500 MeV and 1 GeV does the job).

As an example, let me show the resultfor the low energy constant Lr3(Mρ)Lr3 = LV3 + LS3 = −34G2VM 2V+ 12c2dM 2S= (−3.55 + 0.53) · 10−3(2.47)As advocated, only the resonance masses (MV = 770 MeV, MS = 983 MeV) and cou-plings appear (|GV | = 53 MeV, |cd| = 32 MeV). In table 1, we show the correspondingvalues for all low energy constants.

It is apparent that the resonances almost completelysaturate the Lri , with no need for additional contributions. This method of estimatingLri is sometimes called QCD duality or the QCD version of VMD.

In fact, it is rathernatural that the higher lying hadronic states leave their imprints in the sector of thelight pseudoscalars – as already stated, the typical resonance mass is the scale of newphysics not described by the Godstone bosons.Finally, I mention some other attempts to estimate these low energy constants.These are based on Nambu–Jona-Lasinio models [34], extended Nambu–Jona-Lasinio18

approaches [35], QCD coupled to constituent quarks [36] or solution to Coulomb gaugeQCD in the ladder approximation [37]. All of this give results similar to the proceduredescribed above.

It is important to have such a tool since in many circumstances (q6corrections, non–leptonic weak interactions) the number of coupling constants is toolarge to allow for a systematic phenomenological determination.iLri (Mη)Lri (Mρ)LRESi10.9 ± 0.30.7 ± 0.30.621.7 ± 0.71.3 ± 0.71.23∗−4.4 ± 2.5−4.4 ± 2.5−3.040.0 ± 0.5−0.3 ± 0.50.052.2 ± 0.51.4 ± 0.51.460.0 ± 0.3−0.2 ± 0.30.07∗−0.4 ± 0.15−0.4 ± 0.15−0.381.1 ± 0.30.9 ± 0.30.997.4 ± 0.26.9 ± 0.26.910−5.7 ± 0.3−5.2 ± 0.3−6.0Table 1:Low–energy constants for SU(3)L × SU(3)R. The first two columnsgive the phenomenologically determined values at µ = Mη and µ = Mρ. TheLri (i = 1, .

. ., 8) are from ref.

[14], the L9,10 from ref.[43]. The ’∗’ denotesthe constants which are not renormalized.

The third column shows the esti-mate based on resonance exchange [32]. The constant L10 has recently beenreexamined by Donoghue and Holstein [261].2.6.

ExtensionsHere I will briefly discuss some extensions of the chiral effective Lagrangian atnext–to–leading order. I will be sketchy on these topics and the interested reader shouldconsult the pertinent literature.The first extension concerns the energy–momentum tensor in chiral theories.

LetΘµν denote the energy–momentum tensor. Matrix elements of the type < ππ|Θµµ|0 >appear in the decay of a light Higgs [40,49] (which is by now excluded experimentally)or in the ππ spectrum of the transitions ψ′ →ψππ and Υ′ →Υππ (when one makes useof the multipole expansion [38]).

Donoghue and Leutwyler have enumerated the newterms and performed the pertinent renormalization procedure [39]. There are three newterms which couple the meson fields to the Ricci tensor and the curvature scalar plus19

three contact terms involving squares of the curvature. These have no meson matrixelements.

The resulting energy–momentum tensor to order q4 is given in Ref. [39],Θµν = Θµν2+ Θµν4,g + Θµν4,R + O(q6)(2.48)where Θµν4,g are the conventional terms arising from (2.29) but with Lorentz indicesraised and lowered with gµν and Θµν4,R is generated by the new couplings.The oneparticle matrix elements < p′|Θµν|p > (for π, K, η) are analysed.

They contain twoform factors, a scalar and a tensor since Θµν contains a spin–zero and a spin–two part.Comparison with a dispersive analysis of the scalar form factors [40] allows to pin downLr11 and Lr13, Lr12 is estimated via tensor f2–exchange. The physical significance of thesenew couplings is the following.

While Lr12 determines the slope of the tensor form factor,the slope of the scalar form factor is given by 4Lr11 + Lr12. The combination Lr11 −Lr13measures the flavour asymmetries generated by the quark masses.

In this context, onecan also make contact to the dilaton model of the conformal anomaly [41]. In thatmodel, the breaking of the conformal symmetry is given in terms of an effective scalarfield.

The pertinent low–energy constants are saturated by this scalar and a comparisonto the empirical values of Lr11 and Lr13 shows that the model can be considered semi–quantitative at best. Translating the Lr11 into the mass Mσ, one finds 560 MeV ≤Mσ ≤720 MeV, whereas the empirical value for Lr13 leads to a larger scalar mass, 1000 MeV≤Mσ ≤1100 MeV.The second extension I want to address concerns the axial U(1) transformations.Even in the limit of vanishing quark masses, the U(1)A Noether current is not conservedi∂µJ(0)5µ = 3αs8π G ˜G + 2Xq=u,d,smq ¯qγ5q(2.49)Since G ˜G is a total divergence, one can define a gauge invariant current which generatesU(1)A symmetry transformations in the chiral limit.

A chiral rotation with its associatedcharge ˜Q5 shifts the θ vacuum of QCD, exp{iα ˜Q5}|0 >= |θ −6α > with α the angleof the chiral rotation. One therefore adds a θ source term to the QCD Lagrangian (cf.eq.(2.2)).

In the basis where the quark mass matrix is diagonal QCD without sourcesis characterized by a vacuum angle θ(x) = ¯θ. θ can be added to the chiral Lagrangianin a straightforward manner [14,42].

The lowest order term is already given in eq. (2.9)provided one substitutes the external field χ by ˜χ = χ exp(iθ/3).

The θ–dependenceis then entirely contained in this new source (apart from a contact term). Clearly, ˜χis invariant under global U(1)A rotations.

At next–to–leading order, one has the sameterms as before (with χ →˜χ) and five additional operators involving derivatives of θ plushigher order contact terms with no meson matrix elements. These terms are enumeratedin ref.

[42], for the later applications we only need one of themLDθ4= iL14DµθDµθ Tr (˜χ†U −U † ˜χ) + . .

. (2.50)20

Finally one can also couple a pseudoscalar SU(3)–singlet field like e.g. the η′.

This isspelled out in detail in the original work of Gasser and Leutwyler [14] and will not beneeded in what follows.III. LIGHT QUARK MASS RATIOSThe current quark masses characterize the strength of the flavour symmetry break-ing.

They are small on typical hadronic scales as indicated by the fact that the SU(3)predictions work fairly well.Expanding physical quantities in powers of the quarkmasses, the deviations from the symmetry limit are probing the quark masses. To low-est order, this is a clean method.

The perturbation is of the form ¯qMq and formingratios, one can eliminate the dependence on the unknown operator ¯qq. I will review howCHPT allows to fix the ratios of the light quark masses at leading and next–to–leadingorder.

In the latter case, some model dependence enters as will be discussed later on.For a fairly comprehensive review on the history of the subject and the determinations ofthe absolute values of the quark masses, the reader is referred to Gasser and Leutwyler[3]. Also, as a cautionary remark, let me mention that the mass parameters appearingin the Lagrangian should not be confused with the so–called constituent masses.

Theserepresent the energy of quarks confined inside hadrons. They do not follow directly fromQCD but rather from some effective model.3.1.

Lowest order estimatesThe best place to investigate the light quark mass ratios is the spectrum of thepseudoscalar Goldstone bosons.As already discussed in section 2, their masses aredirectly proportional to the quark masses. Neglecting any electromagnetic and higherorder corrections, the standard GMOR scenario [18] (see also ref.

[44]) givesM 2π+ = (mu + md) B ,M 2K+ = (mu + ms) B ,M 2K0 = (md + ms) B ,(3.1)which allows to extractmumd= 0.66 ,msmd= 20.1 ,ˆmms=124.2(3.2)with ˆm = (mu + md)/2 the average light quark mass. This estimate is, however, some-what too naive.

The charged pions and kaons are surrounded by a photon cloud whichleads to a mass shift of order e2 (isospin breaking). In the chiral limit, one can re-sort to Dashen’s theorem [45] which states that the electromagnetic self-energy of apseudoscalar meson P is proportional to its charge QP ,M 2P = (M 2P )strong + e2QP C + O(e2M)(3.3)21

As shown by Das et al. [46], the constant C can be expressed as an integral over thevector and axial–vector spectral functions.

The latter have been determined from thesemi–leptonic τ →ντ +nπ decays [5]. Another way of fixing this constant is based on theempirical fact that for the pion the mass difference is almost entirely of electromagneticorigin, (Mπ+ −Mπ0)strong ∼(mu −md)2.

This strong splitting is due to the π0η mixingwhich causes the neutral pion to become lighter than its charged partner. Numerically,this effect is tiny, ∼0.2 MeV [3].

Thus, the pion mass difference fixes the value of C,which for the kaons results in the following leading–order strong splitting:(M 2K0 −M 2K+)strong = M 2K0 −M 2K+ + M 2π+ −M 2π0(3.4)which amounts to (MK0 −MK+)strong = (5.3 −1.3) MeV = 4 MeV [3] and modifies thequark mass ratios accordinglymumd= 0.55 ,msmd= 20.1 ,ˆmms=125.9(3.5)This is the celebrated result of Weinberg [47]. Before discussing the O(M2) correctionsto these numbers, let us pause for a moment.

The ratio mu/md is rather different fromone, so why does one not see large isospin violations in physical processes? The answerlies in the fact that although md ≃2mu, both masses are so small compared to thetypical hadronic scale that this effect is almost perfectly screened.

One exception tobe discussed later is the decay η →3π, whose amplitude is proportional to md −mu.Weinberg [47] has also proposed to look into isospin breaking effects in the pion–nucleonscattering lengths, but considering the present status of the empirical information, thisdoes not seem to be a realistic proposal. However, before elaborating further on thesetopics, let us consider the corrections of order M2.3.2.

Next–to–leading order estimatesBefore discussing various attempts to pin down the order M2 corrections to thequark mass ratios (3.5), we have to discuss one subtlety which arises at next–to–leadingorder. As noted by Kaplan and Manohar [48], the chiral symmetry does not differentiatebetween the conventional mass matrix M = diag(mu, md, ms) (with mu,d,s real) or M’defined by the eigenvaluesm′u = α1mu + α2mdms ,m′d = α1md + α2mums ,m′s = α1ms + α2mumd ,(3.6)with α1, α2 two arbitrary constants.

As already noted, only the product BM (via thescalar field s(x)) enters the chiral Lagrangian and thus α1 merely renormalizes the valueof B. The constant α2 enters at order M2 and contributes to L4.

To be more specific,22

it contributes to the terms with two powers of χ. One can therefore completely hidethis symmetry by redefiningB′ = B/α1 , L′6 = L6 −c , L′7 = L7 −c , L′8 = L8 + 2c .

(3.7)with c = (α2/α1) (F 2/32B). This means that L2 + L4 is invariant under the symmetry(3.7).

Notice that the term proportional to Lr5 is not affected since it only containsone power of the quark mass matrix. Furthermore, it should be stressed that this isnot a symmtery of QCD but rather an artefact of the truncated chiral expansion (fora different view, see Ref.[42]).

In any case, some theoretical arguments are necessaryto overcome this ambiguity if one wants to extract the corrections of order M2 to thequark mass ratios.Let us consider first the classical analysis of Gasser and Leutwyler [14]. The chiralexpansion of the pseudoscalar masses at next–to–leading order readsM 2π =◦M2π1 + µπ −13µη + 2 ˆmK3 + K4,M 2K =◦M2K1 + 23µη + ( ˆm + ms)K3 + K4,M 2η =◦M2η1 + 2µK −43µη + 23( ˆm + ms)K3 + K4+ K5 +◦M2π−µπ + 23µK + 13µη.

(3.8)withµP =M 2P16π2F 2 ln(MP/µ) ,K3 = 2κ(2Lr8 −Lr5) , K4 = 4(2 ˆm + ms)κ(2Lr6 −Lr4) ,K5 = 89κ2(3L7 + Lr8) , K6 = κLr5/F ,K7 = 2κ(2 ˆm + ms)Lr4/F , κ = 4B/F . (3.9)neglecting terms of the order mu −md.

Inspired by the discussion of the lowest ordermass ratios, one forms the dimensionless quantitiesQ1 = M 2KM 2π= ˆm + ms2 ˆm(1 + ∆)Q2 = (M 2K0 −M 2K+)strongM 2K −M 2π= md −mums −ˆm (1 + ∆) . (3.10)Remarkably, in both cases the correction ∆is the same ( for an explicit expression, seeref.[14]).

Therefore, up to corrections of order M2 one can form a ratio independent ofthe low energy constants (which are hidden in the explicit value of ∆):mumd2 + Q2Q1msmd2 = 1(3.11)23

which defines an ellipsis with semi–axis 1 andpQ2/Q1, respectively. Making use ofDashen’s theorem, this leads topQ2/Q1 = 23.6.

Therefore, eq. (3.11) constitutes a low–energy theorem – any value of the mass ratios mu/md and ms/md must fulfill this ellipticconstraint (modulo corrections to Dashen’s theorem, see below).

However, the actualvalues of these ratios depend on the value of the correction ∆. This depends on the low–energy constants Lr5 and Lr8.

Redefinitions of Lr5,8 allow one to move along the ellipse.In particular, one can choose mu = 0 which is favored in connection with the strong CP-problem [50] (see also the instanton gas calculation reported in ref.[51]). However, thereare strong phenomenological constraints on the low–energy constants.

First, the value ofLr5 is rather narrowly pinned down by the coupling constant ratio FK/Fπ = 1.22 ± 0.01[52] leading to the value given in table 1 (it is also consistent with scalar resonanceexchange for MS ≃1 GeV). Concerning Lr8, the situation is less favorable.

Making useof the deviation from the Gell-Mann–Okubo relation (cf. eq.

(3.8)), which involves thecouplings Lr5, L7 and Lr8 and is well known, Leutwyler [53] has proposed to estimate L7via η′ exchange as given in eq(2.38). L7 also enters the determination of the ηη′ mixingangle, which is empirically constrained to Θηη′ = (22 ± 4)◦[54].

This allows to fix Lr8.The elliptic constraint together with the one from the ηη′ mixing leads tomumd= 0.55 ± 0.12 , msmd= 20 ± 2.2 ,(3.12)rather consistent with Weinberg’s estimate (3.5). A further check comes from the baryonmass spectrum.Gasser [55] and Gasser and Leutwyler [3] have analyzed the next–to–leadig order quark mass corrections which scale as M3/2.

Combining the isospinbreaking mass splittings mp −mn, mΣ+ −mΣ−and mΞ0 −mΞ−one arrives atR = ms −ˆmmd −mu= 43.5 ± 3.2(3.13)This result for R is also supported by examining the ρω splitting at next–to-leadingorder [3]. Imposing this constraint on the results (3.12), one ends up withmumd= 0.56 ± 0.06 ,msmd= 20 ± 2 ,ˆmms=125.6 ± 2.0(3.14)which shows that the meson spectrum (consistently with the baryon mass splittings)determines the quark mass ratios rather accurately.

This is a consistent picture. How-ever, what is definitively missing is a better treatment of the corrections to Dashen’stheorem.Donoghue and Wyler [42] have taken a somewhat different path to get a handleon the second order corrections to the quark mass ratios.

They were inspired by theinstanton gas calculation of Choi et al. [51].

In that calculation, in which an originallymassless up quark travels through the instanton gas, it acquires an effective mass,meffu ∼mdmseiθ(3.15)24

due to quantum effects (’t Hooft’s six–fermion term which has been shown to be ofutmost importance in understanding the problem of flavor mixing in theories with quarkdegrees of freedom [56]).Notice that the vacuum angle θ enters the game.The θ-dependence is the main ingredient in the calculation of ref. [42] to overcome the Kaplan-Manohar ambiguity.

The basic observation is that the operator G ˜G, which follows fromLQCD after variation by θ, is immune to the reparametrization invariance of the quarkmasses. However, since one can not directly measure the θ-dependence from strong CP-violating effects, one has to resort to the multipole expansion for heavy quark systems[38].

The decays V →V + M with V = Ψ or Υ and M = π0, η, 3π are determinedby the operator G ˜G to leading order in the heavy quark expansion. Taking ratios, onecan eliminate almost all model dependence.

The strength of G ˜G can be parametrizedin terms of the number rG ˜G,rG ˜G = < 0|G ˜G|π0 >< 0|G ˜G|η >= 3√34 (md −mums −ˆm ) FηFπ×1 −32BF 2π(ms −ˆm)(L7 + Lr8) + 4Lr14F 2π(M 2π −M 2η ) −(3µπ −2µK −µη)(3.16)To arrive at this result, one uses ∂Leff/∂θ ∼G ˜G and sandwiches the operator betweenthe pertinent states. Clearly, all the terms in Leffwhich contain ˜χ = χ exp(iθ/3) cancontribute and there is also the term proportional to Lr14 (cf.

eq. (2.50)) which is linearin θ.

The first term on the right hand side of (3.16) agrees with the result of Novikov etal. [57].

One can furthermore form one particular combination of the mass ratios whichis free of chiral logarithms (contained in the µP ) and only depends on one low–energyconstant [42]md −mumd + mums + ˆmms −ˆm = 4(F 2KM 2K −F 2πM 2π)3√3F 2πM 2πFπFηrG ˜G[1−∆GMO]1+ 4Lr14Fπ2 (M 2η −M 2π)(3.17)so that one has to determine rG ˜G and Lr14 to get this mass ratio. The first quantity canbe determined fromΓ(V ′ →V π0)Γ(V ′ →V η) = r2G ˜Gpπpη3(3.18)with pπ(pη) the momentum of the π(η).

The data on the decays Ψ′ →J/Ψ + π0 andΨ′ →J/Ψ + η lead to rG ˜G = 0.043 ± 0.0055. Furthermore, one can use the principle ofresonance saturation to estimate Lr14.

The authors of ref. [42] give the following limits:0 ≤4Lr14/F 2π ≤M 2η′.

Combining pieces and adding the errors (theoretical and empirical)in quadrature, this leads tomd −mumd + mums + ˆmms −ˆm = 0.59 ± 0.11(3.19)25

or equivalently (since ms ≫ˆm):mumd= 0.30 ± 0.07 . (3.20)This result gives further credit to the notion that mu is indeed unequal zero, but itsrather large deviation to the value given in (3.14) should be noted.

Such an effect canonly occur if there are substantial corrections to Dashen’s theorem, much bigger thanassumed in the calculation of Leutwyler [53]. This will be discussed below.

The estimateof Donoghue and Wyler can be criticied on two grounds. First, it corresponds to a valueof L7 which is not in particular good agreement with phenomenology.

Second, it is notclear that the multipole expansion is justified in the Ψ′(Ψ) system. A measurement forthe upsilon system would be very much needed to calrify the uncertainties inflicted bythe heavy quark expansion on the result (3.20).Donoghue, Holstein and Wyler [58] have extended this latter analysis by consideringalso the η →3π decay and a model to estimate the corrections to Dashen’s theorem(see also ref.

[59] for an earlier attempt). They consider the particular mass ratioR1 = md −mums −ˆm2 ˆmms + ˆm = MπMK2 (M 2K0 −M 2K+)strongM 2K −M 2π(3.21)which is related to the strong kaon mass difference and also appears in the chiral expan-sion of the decay amplitude η →π+π0π−.

The latter has been worked out to one looporder by Gasser and Leutwyler [60]. The matrix element for η →3π can be parametrizedby one single function of the pertinent Mandelstam variables, denominated A(s, t, u).To leading order, the chiral expansion of A(s, t, u) readsA(s, t, u) = −(md −mu)B3√3F 2π{1 + 3(s −s0)M 2η −M 2π+ O(q2) + O(e2 ms −ˆmmu −md)}(3.22)with s0 = M 2π + M 2η/3 the center of the Dalitz plot.

While current algebra (tree level)predicts Γ(η →π+π0π−) = 66 eV, the one–loop calculation leads to (160 ± 50) eV (theuncertainty is an estimate of the higher order terms in the chiral expansion). Noticethat these values are in disagreement with the empirical result of (281 ± 29) eV.

Thelarge enhancement of the one–loop result over the tree level prediction is essentially anunitarity effect, i.e. an effect of the strong pionic final state interactions in the S-wave.This was originally predicted by Roiesnel and Truong [15] who found an even larger dis-crepancy (see the discussion in ref.[60]).

Notice also that these uncertainties essentiallydrop out in the ratio Γ(η →3π0)/Γ(η →π+π0π−) = 1.51, 1.43 for current algebra andone loop CHPT [60], respectively. The emiprical value is 1.35 ± 0.04, comparable tothe one loop prediction.

The authors of ref. [58], however, turn the argument around.They use the empirical rate for η →π+π0π−to fix the strong kaon mass difference sinceΓ(η →π+π0π−) fixes R1 (3.21).

This gives(∆M 2K)strong = (M 2K0 −M 2K+)strong = 2MK · (7.0 MeV) . (3.23)26

This is considerably different from the result based on Dashen’s theorem, (∆M 2K)strong =2MK · (5.3 MeV) (a 30 per cent deviation). To further support this argument, they alsoconsider a model to independently estimate the corrections to Dashen’s theorem.

It isbased on the empirical observation that vector meson dominated form factors in the Borndiagrams give a rather accurate description of the electromagnetic pion mass difference[16,32,33] (see fig.3). Beyond leading order and making use of the Weinberg sum rules(which is necessary to cancel the divergent pieces) this leads toFig.

3:Electromagnetic self–energy of pseudoscalars to order e2 in the reso-nance exchange picture. The wiggly lines denote photons, the double solid linesvector or axial mesons.

Tadpole diagrams are not shown.∆M 2π = 2Mπ · (6.3 MeV)(3.24)using the physical mass of the A1 meson. The result (3.24) is considerably larger thanthe empirical value ∆M 2π = 2Mπ ·(4.6 MeV).

The lowest order result, however, is withina few per cent of the empirical value, ∆M 2π = 2Mπ · (4.7 MeV) [32]. This sheds somedoubt on the accuracy of this approach.

To leading order, the formalism used to arriveat (3.24) respects Dashen’s theorem but predicts a strong violation of it at next–to–leading order, (∆M 2K)em = 1.78 (∆M 2π)em. This large deviation from one is essentiallydue to the kaon propagator in the Born terms (cf.

fig.3) and thus the strong kaon massdifference is affected,(∆M 2K)strong = 2MK · (6.3 ± 0.1 MeV)(3.25)which agrees within 10 per cent with the number (3.23) extracted from the η →3π decay.Assuming furthermore resonance saturation for the low–energy constants L7 and Lr14(η′ exchange), the authors of ref. [58] can also determine the mass ratios 2 ˆm/( ˆm + ms)27

and (md −mu)(ms + ˆm)/(md + mu)(ms + ˆm) (from the meson masses and the decaysΨ′ →J/Ψ + π, η) and arrive atˆmms= 131 , md −mums= 129 , md −mumd + mu= 0.59 ,(3.26)ormumd= 0.26 . (3.27)This value for the ratio of the up and down quark masses is consistent with the onegiven in (3.20).

While this set of mass ratios has the virtue of giving a correct η →3πamplitude, which is a notorious problem in CHPT, there are various aspects whichhave to be looked at in more detail. First, from (3.26) one deduces R ≃28 which isfour standard deviations offthe value extracted from the baryon spectrum (see also thediscussion at the end of this section).

Furthermore, it is not obvious that the one–loopresult for the amplitude A(η →3π) should be used for fixing the mass ratio R1. As wewill see from the discussion of the scalar form factor in section 4, strong pionic final stateinteractions cause the failure of the one loop approximation close to two–pion threshold.For η →3π, one deals with √s ≈550 MeV.

Finally, the large next–to–leading ordercorrections to the pion mass difference put a question mark on the convergence of thisapproach.From the discussion so far it should have become clear that the calculations ofthe quark mass ratios at next–to–leading order involve some model-dependence. It istherefore necessary to investigate as many constraints as possible (meson masses, baryonmasses, η →3π, corrections to Dashen’s theorem and alike) to arrive at a consistentpicture of the quark mass ratios at order M2.

If the modifications to mu/md and ˆm/msare indeed as large as indicated by (3.20) or (3.26), it is not obvious why one should stopat next–to–leading order. The set of quark mass ratios (3.14) has the virtue of showingstability against inclusion of the quark masses as perturbations.

At present, I considerit as the most consistent estimation of the light quark mass ratios.There is some additional evidence which gives further credit to the constraint fromthe baryon sector, i.e. the value of R (3.13).

This work is based on so-called secondorder ”flavor” perturbation theory [61].It extends the work of Gell-Mann, Okuboand others [17]. It differs from CHPT in that the flavor symmetric part of the quarkmass term (mu + md + ms)(¯uu + ¯dd + ¯ss)/3 is already contained in the unperturbedHamiltonian.

The basic idea is that in this way one includes already most of the (large)M3/2 corrections of CHPT and thus deals with smaller perturbations. If one works outthe flavor perturbations to second order, one finds [61]:R =3mΛ + mΣ −2mN −2mΞ2√3mT + (mn −mp) + (mΞ0 −mΞ−) .

(3.28)Here, mT is the Λ–Σ0 transition mass. Its physical effect is that it produces a smallmixing ǫ of the mass–diagonal fields, ǫ = mT /(mΣ −mΛ).

Empirically, it is not well28

determined. It could be pinned down from a precision measurement of the differencein the pK−→Λη and n ¯K0 →Λη cross sections.

At present, one can only deduce a(conservative) lower limit of R ≥38 ± 11. A similar result can also be obtained fromthe splittings in the decuplet.

Finally, if one assumes that the first order result lies asclose as possible to the family of second order solutions, one finds R = 48 ± 5, quiteconsistent with the value given in eq.(3.13). It is certainly desirable to have such extrainformation to strengthen our understanding of the ratios of the light quark masses.Some arguments which seem to support the notion of a much smaller ratio ms/ ˆm arediscussed in section 4.2.IV.

THE MESON SECTOR – SELECTED TOPICSIn this section, I discuss a few selected topics of applying CHPT in the mesonsector.Clearly, space forbids to account for less than a small fraction of the manyexisting predictions and their comparison to the data. In section 8, additional referencesconcerning some of the neglected topics are given.Here, I will first discuss the classical topic of pion–pion scattering within the oneloop approximation.

This is an important topic since, as will be discussed, it allows todirectly test our understanding of the mechanism of quark condensation in the vacuum.I also discuss briefly the related subject of πK scattering. Then, the question of therange of applicability of the chiral expansion is discussed.

Various schemes to extend theenergy range are critically reviewed. I also sketch the extenstion of the chiral Lagrangianat next–to–leading order to include the non–leptonic weak interactions with particularemphasis on the decay modes K →2π and K →3π.4.1.

ππ scatteringThe scattering of pions is in a sense the purest reaction to test our understanding ofthe low energy sector of QCD. It involves only the pseudoscalar Goldstone bosons andtheir dynamics.

Also, for the reaction ππ →ππ one can restrict oneself to the sectorof SU(2) × SU(2), where the quark mass corrections are expected to be very small.To be more specific, consider the process πa(pa) + πb(pb) →πc(pc) + πd(pd), for pionsof isospin ’a, b, c, d’ and momenta pa,b,c,d. The conventional Mandelstam variables aredefined via s = (pa + pb)2, t = (pa −pc)2 and u = (pa −pd)2 subject to the constraints + t + u = 4M 2π.The scattering amplitudes can be expressed in terms of a singlefunction, denoted A(s, t, u):T cd;ab(s, t, u) = A(s, t, u) δabδcd + A(t, s, u) δacδbd + A(u, t, s) δadδbc(4.1)for the two flavor case.

The chiral expansion of A(s, t, u) takes the formA(s, t, u) = A(2)(s, t, u) + A(4)(s, t, u) + O(q6) ,(4.2)29

where A(m) is of order qm and the symbol O(q6) denotes terms like s3, s2t, st2, . .

.. Theexplicit form of the tree level amplitude was first given by Weinberg [62]. It can beeasily read offfrom eq.

(2.9) by expanding in powers of the pion field and collecting theterms proportional to π4,A(2)(s, t, u) = s −M 2πF 2π+ O(q4) ,(4.3)where it is legitimate to use the physical values of the pion mass and decay constantsince the differences to their lowest order values is of order q4. The next-to–leading orderterm has been worked out by Gasser and Leutwyler [13,63] (cf.

fig.2). One splits it intounitarity corrections (denoted B(s, t, u)) and tree and tadpole contributions (denotedC(s, t, u)).

The explicit form reads (I use the one given in ref. [64]):A(4)(s, t, u) =B(s, t, u) + C(s, t, u)B(s, t, u) = 16F 4π3(s −M 4π) ¯J(s) +[t(t −u) −2M 2πt + 4M 2πu −2M 4π] ¯J(t) + (t ↔u)C(s, t, u) =196π2F 4π2(¯ℓ1 −4/3)(s −2M 2π)2 + (¯ℓ2 −5/6)[s2 + (t −u)2]+ 12M 2πs(¯ℓ4 −1) −3M 4π(¯ℓ3 + 4¯ℓ4 −5)¯J(z) = −116π2Z 10dx ln[1 −zx(x −1)/M 2π] .

(4.4)In the chiral limit, one recovers the form of A(s, t, u) first given by Lehmann [65].Unitarity and analyticity force the appearance of the loop terms which contain twounknown scales. These can be expressed in terms of the low energy constants ¯ℓ1,2 [13].The polynomial term C(s, t, u) contains two further constants related to the shift of Fπand Mπ away from their lowest order values.

An explicit expression of A(4)(s, t, u) interms of the SU(3) representation can be found in ref. [66].For comparison with the data, one decomposes T cd;ab into amplitudes of definitetotal isospin (I = 0, 1, 2) and projects out partial–wave amplitudes T Il (s),T Il (s) =p1 −4M 2π/s2iexp2i[δIl (s) + iηIl (s)]−1(4.5)with s = 4(M 2π + q2) in the c.m system, l denotes the total angular momentum of thetwo–pion system.

The phase shifts δIl (s) are real and the inelasticities ηIl (s) set in atfour–pion threshold. They are, however, negligible below ¯KK threshold (below whichthey are a three–loop effect of order q8).

Near threshold (s = 4M 2π) the partial–waveamplitudes take the formRe T Il = q2l{aIl + q2 bIl + O(q4)} . (4.6)30

The coefficients aIl are called scattering lengths, the bIl are the range parameters. It wasalready observed by Weinberg [62] that the scattering lengths resulting from the treelevel were much smaller than naively expected.

This is also borne out in the one loopcalculation [13]. Let me now concentrate on the isospin–zero S–wave scattering lengtha00.

The tree level prediction is a00 = 7M 2π/32πF 2π = 0.16 [62] and to one loop order, onecan derive the following low energy theorem [13],a00 = 7M 2π32πF 2π1+ 13M 2π < r2 >πS −M 2π672π2F 2π(15¯ℓ3−353)+ 254 M 4π(a02+2a22)+O(q6)(4.7)which involves the scalar radius of the pion, < r2 >πS≃0.7 fm2 (see also section 4.4),the low energy constant ¯ℓ3 and the D–wave scattering lenghts a02 and a22. Taking thelatter from Petersen [67] and allowing for conservative errors on the scalar radius and¯ℓ3, one arrives at [13,63]a00 = 0.20 ± 0.01 .

(4.8)One makes two important observations. First, there is a 25 per cent correction to thetree result (which is due to the fact that the non–analytic term of the type M 2π ln M 2πhas a large coefficient).Second, the theoretical uncertainty is rather small.

This ismostly due to the fact that the poorly known ¯ℓ3 is multiplied by a tiny factor. We willcome back to this important result in section 4.2.

For a more detailed account of thethreshold behaviour of the other partial waves, see refs. [13,63].If one now increases the energy, there are two main issues to address.

First, howdoes the truncated chiral series (4.2) compare with the data and, second, at what energydo the one–loop corrections become so large that one loop approximation can not betrusted any more? Before addressing these issues, I have to discuss an ambiguity whicharises in the extraction of the phase shifts from eq.(3.6).

One can either use [64]δIl = Re T I(2)l+ Re T I(4)l+ O(q6)(4.9)orδIl = arctan (Re T Il ) + O(q6) . (4.10)The difference between these two forms only shows up at order q6 and therefore givesan estimate about the relative importance of these higher order terms (in what follows,I will mostly use the definition (4.9)).

In fig.4a,b,c the phase shifts δ00(s), δ11(s) andδ20(s) are shown, respectively [64]. For comparison, the tree result and the existing dataare shown together with a band of Roy equation fits [68,69] (the latter impose unitarityand analyticity and, obviously, some of the data are in conflict with these fundamentalprinciples).

At the time the Roy equation program was carried out [67,69], the S–wavescattering length, which is a fundamental input, was not known. The band indicatedin the figures corresponds to 0.17 ≤a00 ≤0.30.

Before discussing the results, let mepoint out that for δ00 one has additional information in the threshold region from Kℓ431

Fig. 4:ππ scattering phase shifts δ00 (a), δ11 (b) and δ20 (c), in order.

Thedashed line gives the tree result and the dashed–dotted the one–loop prediction.Also shown is the Roy equation band as discussed in the text. The data canbe traced back from ref.[64].

In (a), the double–dashed line corresponds to theone–loop result based on eq. (4.10).32

decays.The energy where the one–loop contribution δI(4)lis half as big as the treelevel prediction (in magnitude) is indicated by the shaded vertical line. Notice thatthis critical energy is different for the various channels, √s = 450 MeV, 480 MeV and470 MeV, in order.Beyond these energies, the truncated chiral expansion becomesunreliable.

In the isospin–zero S–wave (fig.4a), the data below √s = 600 MeV are poor(with the exception of the ones from Kℓ4 decays). Notice that both representations(4.9) and (4.10) stay within the Roy equation band up to the critical energy.

Below 400MeV, they are virtually identical. In the P–wave, the well–known ρ resonance shrinksthe Roy equation band to a narrow line.

The chiral prediction follows it up to 500 MeV.Clearly, it is not possible to describe the resonance behaviour by this method. This isthe natural barrier to the theory of pseudoscalars only already mentioned before.

Inthe exotic S–wave (I = 2), the tree and one–loop contributions are of opposite sign.This leads to a stronger sensitivity of the chiral predictions to the actual values of thelow–energy constants (see the shaded area in fig.4c, which corresponds to a change ofℓ4 by one unit. The central value of ℓ4 gives the lower rim of this area.

)It was noted in ref. [64] that the phase of the parameter ǫ′, which measures di-rect CP–violation in the kaon system, can nevertheless be extracted with rather goodaccuracy (see also refs.[70,71]).

It is defined via [72]Φ(ǫ′) = (δ20 −δ00)s=M2K0+ π2(4.11)(see also section 4.5). One therefore needs the S–wave phase shifts at √s = MK0 = 494MeV, i.e.

at an energy where the one–loop coorections are already large. However, inthe difference these corrections largely cancel and one findsΦ(ǫ′) = (45 ± 6)◦,(4.12)to be compared with the tree result of 53◦.

The uncertainty of 6◦is a combination of theuncertainties in the low energy constants and the higher order contributions. Ochs [73]has recently reanalyzed results of phase shifts from π−p →nπ−π+ for energies above600 MeV.

Using Roy equation methods, he finds Φ(ǫ′) = (43 ± 6)◦, in good agreemnetwith the chiral prediction. This is an important result since it essentially pins down thisparameter.

Attempts to extract Φ(ǫ′) from the various K →2π modes are hamperedby the fact that there are large corrections from isospin breaking which have not yetbeen investigated in full detail (some references can be traced back from ref. [64]).Donoghue et al.

[74] have taken a different point of view on the one–loop results.They used the available data on the S, P, D partial–wave amplitudes in the range ofenergies from threshold to 1 GeV to determine the low energy constants ¯ℓ1,2 by a bestglobal fit to these data. This leads to values which are somewhat different form thestandard ones [13,14].

However, from the discussion before it should be clear that sucha fit is not quite legitimate since the one–loop corrections are too large beyond 50033

MeV. While the authors of ref.

[74] find a generally satisfactory description of the dataup 700 . .

.800 MeV, it is not clear what one learns from such a comparison. Notice alsothat the partial–wave amplitudes are bounded in magnitude by unitarity and thereforea much less sensitive probe of deviations than the phase shifts.There is furthermore a whole set of articles in which the one–loop results are ex-tended to higher energies based on unitarization schemes or explicit resonance exchanges.These proposals will be discussed in section 4.4.

Let me now return to the importanceof the S–wave scattering lengths.4.2. Testing the mode of quark condensationSo far, we have assumed the standard scenario in which < 0|¯uu|0 > is the orderparameter of the spontaneous chiral symmetry breaking and the constant B = −<0|¯uu|0 > /F 2 is so large (∼1 GeV) that the first term in the quark mass expansionof the pseudoscalar masses (BM) dominates (this is also the point of view I subscribe,it is backed by lots of phenomenological and some theoretical arguments, see below).However, this notion has been challenged already in the seventies [75,76].

It was claimedthat the possibility M 2P ∼M2 fits the data as well (in that case, B = 0 in the chirallimit). Some recent work along these lines has been done by Fuchs et al.

[77], whoargued that the Goldberger–Treiman discrepancies in the baryon octet (the deviationsfrom the GT relations, which are exact in the chiral limit) lead to an upper bound forthe quark mass ratio ms/ ˆm ≤10.6±4.2. This is more than 3 standard deviations offthestandard value (3.14) (for earlier work on this problem, see the review by Dominguez[78]).

The same authors have recently proposed a modified CHPT [79] to account forthis. Their starting point is the claim that a scenario in which B is small, B ∼ΛQCD ∼Fπ ∼100 .

. .200 MeV is as natural as the large B case.

Then, the ratio ms/B is oforder one and one has to modify the chiral power counting.Apart from the smallexternal momenta, one has to consider the quark mass matrix M and the parameter Bsimultaneously as small quantities, with the ratio ms/B kept fixed. It is then natural toassign the external field χ the dimension one.

The effective Lagrangian takes the form:Leff=Xl,m,nBl Llm,n =XNL(N) ,(4.13)with m covariant derivatives, n quark mass insertions and l factors of B so that l + m +n = N (with l, m, n integers). In the standard case, m + 2n = N. This modifies theleading term of order q2 and new terms at order q3 appear (for explicit expressions, seeref.[79]).

The leading non–analytic contributions are still of order q4, i.e. at next–to–next–to leading order in the modified chiral expansion.The best place to test this scenario is indeed ππ scattering in the threshold region.Remember that the S–wave scattering lengths vanish in the chiral limit and thereforedirectly measure the symmetry breaking of QCD.

This is why these scattering lengths are34

of such fundamental importance. In the modified CHPT, the ππ scattering amplitudereads (up to and including terms of order q3)A(s, t, u) = α M 2π3F 2π+ β s −4M 2π/3F 2π.

(4.14)In the standard scenario, α = β = 1 and one recovers eq.(4.3). In the extreme case thatB vanishes in the chiral limit, one has α = 4 and β = 1.

In general, one can interpolatebetween these two cases via (to leading order)α(2) = 1 + 6(2M 2K/M 2π −1) −(ms/ ˆm)(ms/ ˆm)2 −1(1 + 2ξ) ; β(2) = 1 ,(4.15)where ξ accounts for the Zweig rule violation in the 0++ channel and is expected tobe small (ξ = 0 in what follows). For the value ms/ ˆm = 10 advocated in ref.

[77], onehas α(2) = 1.96. This leads to a00 = 0.19 and a20 = −0.031 to be compared with theWeinberg result of a00 = 0.16 and a20 = −0.045.

The effect is maximal in the combinationa00 + 2a20 [79]. However, one also has to know the corrections of order q3 and q4 to thesepredictions.

They have not yet been worked out in detail. The terms of order q3 leadto a small increase in β and the non–analytic contributions are not known (see belowfor an estimate).

It is, however, clear that this scenario could lead to large value ofa00, say a00 = 0.27 (ms/ ˆm = 6 at tree level). Such a value could not be accomodatedby the standard scenario, cf.

eq.(4.8). At present, the data are not accurate enough todifferentiate between these two cases.

Pion production experiments as recently analyzedby Ochs [73] lead to a00 = 0.23 ± 0.08 and the Kℓ4 data of Rosselet et al. [80] tiedtogther with Roy equation constraints lead to a00 = 0.26 ± 0.05 [67].

Therefore, a betterdetermination of the S–wave scattering lengths is urgently called for. In fact, recentwork at Serpuhkov [81] has established the existence of short-lived π+π−atoms.

In casethey decay from the ground state, their life time is directly proportional to (a00 −a20)2[82]. Therefore, a measurement of the life time with an accuracy of 10 per cent would pindown this combination of the S–wave scattering lenghts within 5 per cent.

A letter ofintent for such an experiment at CERN has been presented [83]. In case this experimentis approved and reaches the projected accuracy, it will give strong indications aboutwhich value of B is favored.

Also, from the Φ factory DAΦNE at Frascati one will getvery much improved statistics on Kℓ4 decays and thus better information on the phaseδ00 in the threshold region.Finally, we should mention that Crewther [84] has already investigated this problemsome time ago. He allowed for a general case M 2π ∼ˆmd, with d an integer.

d = 1gives the standard scenario and d = 2 the extreme small B case. Taking into accountthe leading non–analytic terms (chiral logarithms) with a scale of µ = 800+800−400 MeV,he compared the predictions for the ππ threshold parameters with the available data.Although neither d = 1 nor d = 2 describes the data very well, a slight preference forthe standard case is found.

This result should only be considered indicative, a complete35

one–loop calculation in the modified CHPT scheme has to be performed so that onehas predictions of comparable accuracy to the ones already existing for the standardcase (like eq.(4.8)). It is my believe that when the dust settles, the small B alternativewill be excluded.

Also, the Gell-Mann–Okubo relation for the pseudoscalar masses doesnot follow naturally in the modified scheme, it requires some parameter fitting. Thisis a rather unappealling feature.

Furthermore, there are indications from lattice gaugecalculations supporting the GMOR scenario [85].4.3. πK scatteringThe scattering process πK →πK is of particular interest because it is the sim-plest reaction of the Goldstone bosons involving strangeness and unequal meson (quark)masses. Furthermore, since the low energy constants are already determined from otherprocesses, πK scattering can serve as a test of the large Nc predictions for some of thesecouplings.

It also gives an idea about our understanding of the symmetry breaking inthe strange sector. Obviously, corrections to the tree results are expected to be largerthan in the two flavor case since the small expansion parameter (Mπ/4πFπ)2 = 0.014 issubstituted by (MK/4πFπ)2 = 0.18.First, some kinematics has to be discussed.

In the s channel, there are two inde-pendent amplitudes with total isospin I = 1/2 and I = 3/2. The latter is given by thespecific process π+K+ →π+K+, i.e.T 3/2πK (s, t, u) = T(π+(p1)K+(p2) →π+(p3)K+(p4)) ,(4.16)with s + t + u = 2(M 2π + M 2K).

Using crossing symmetry, the isospin–1/2 amplitudefollows via [86]T 1/2πK (s, t, u) = 32T 3/2πK (u, t, s) −12T 3/2πK (s, t, u) . (4.17)The one–loop result for T 3/2πK (s, t, u) has been given in refs.

[66,86],T 3/2(s, t, u) = T2(s, t, u) + T T4 (s, t, u) + T P4 (s, t, u) + T U4 (s, t, u)T2(s, t, u) =12F 2π(M 2π + M 2K −s)T T4 (s, t, u) =116F 2πµπ[10s −7M 2π −13M 2K] + µK[2M 2π + 6M 2K −4s]+µη[5M 2π + 7M 2K −6s]T P4 (s, t, u) =2F 2πF 2K4Lr1(t −2M 2π)(t −2M 2K)+2Lr2(s −M 2π −M 2K)2 + (u −M 2π −M 2K)2+Lr3(u −M 2π −M 2K)2 + (t −2M 2π)(t −2M 2K)+4Lr4t(M 2π + M 2K) −4M 2πM 2K+2Lr5M 2π(M 2π −M 2K −s) + 8(2Lr6 + Lr8)M 2πM 2K36

T U4 (s, t, u) =14F 2πF 2Kt(u −s)2M rππ(t) + M 2KK(t)+ 32 [(s −t) {LπK(u) + LKη(u)−u(M rπK(u) + M rKη(u))+ (M 2K −M 2π)(M rπK(u) + M rKη(u))+12(M 2K −M 2π)KπK(u)(5u −2M 2π −2M 2K) + KKη(u)(3u −2M 2π −2M 2K)+JrπK(s)(s −M 2π −M 2K)2 + 18JrπK(u)11u2 −12u(M 2π + M 2K) + 4(M 2π + M 2K)2+38JrKη(u)u −23(M 2π + M 2K)2+ 12Jrππ(t) t (2t −M 2π)+34JrKK(t)t2 + 12Jrηη(t)M 2π(t −89M 2K)(4.18)The explicit form of the functions M rP Q and JrP Q is given in [14]. The tree level resultagrees with the one of Weinberg [62] and Griffith [87].

Notice that the amplitude de-pendes, like the ππ amplitude (4.4), on the low energy constants Lr1,2,3,4 but also onLr5,6,8.Fig. 5:Theoretical predictions and empirical data on the S–wave scatteringlengths.

We show the current algebra point (CA) and the one–loop chiral per-turbation theory result (CHPT). The data can be traced back from ref.

[66].In fig.5, the one–loop prediction for the S–wave scattering lengths a1/20and a3/2037

is shown in comparison with the data (see ref. [66]), the result of a Roy equation study[88] and a dispersive analysis [89].

The uncertainty in the prediction comes from theuncertainties of the Lri added in quadrature (which is a conservative estimate). It isclear from that figure that a better determination of these scattering lengths would bevery much needed.

At energies above threshold, the bulk of the data for the phase shiftδ1/20(s) is described fairly well up to 850 MeV (at 800 MeV, the one-loop contributionis half as large as the one from the tree level). For the I = 3/2 phase, only few veryinaccurate data exist in the threshold region.

Similar to the I = 2 ππ S–wave, the treeand one–loop contributions cancel leading to a strong sensitivity to the values of the lowenergy constants. In the P–wave, the nearby K∗(892) resonance forces the one–loopprediction to fail rather quickly.

It should be pointed out that the threshold of πKscattering is at 633 MeV and therefore the loop corrections can become large ratherquickly. It was, however, stressed in ref.

[66] that one can continue the amplitudes intothe unphysical region and expand around the point ν = (s −u)/MK = t = 0. Thisleads to a satisfactory agreement with existing dispersion–theoretical results [90].

Also,the isospin–even amplitude T (+)(ν, t) is most sensitive to the actual values of the Lri . Abetter determination of it would allow to test some of the large Nc predictions for thesecouplings.

In any case, a more accurate determination of the πK scattering amplitudeswould serve as a good test of our understanding of the symmetry breaking in the SU(3)sector of QCD.4.4. Beyond one loopUp to now, we have considered CHPT calculations in the one–loop approximation.In the two flavor case, higher order corrections are generally small at low energies becausethe expansion parameters are M 2π/16π2F 2π = 0.014 and E2pion/F 2π.

Furthermore, theamount of work to perform a multi–loop calculation is substantial and no systematicstudy of the accompanying low energy constants is available. As particularly stressed byTruong and collaborators [91,92] pions in the isospin–zero S–state produce strong finalstate interactions.

As a result of this, the one–loop approximation to e.g. the scalarform factor of the pion or to the phase shift δ00 becomes inaccurate at surprisingly lowenergies.The simplest object to study these effects is the scalar form factor (ff) of the pion,< πa(p′)πb(p) out| ˆm(¯uu + ¯dd)|0 >= δab ˜Γπ(s) ,(4.19)with s = (p′ + p)2.It is not directly measurable, but has been determined for en-ergies from threshold to 1 GeV by a dispersive analysis in ref.

[40] via solution of theMuskhelishvili-Omn`es equations of the ππ/K ¯K system (see also [49]). In ref.

[93], anexplicit calculation of the scalar ffbeyond one loop was performed. Consider the nor-malized scalar ff, Γπ(s) = ˜Γπ(s)/˜Γπ(0).

This quantity is of order one to tree and of orderq2 at one–loop level. To perform the two–loop calculation, one considers the unitarityrelation depicted in fig.6.

The key point is that if one has the imaginary part at order38

Fig. 6:Unitarity relation for the scalar form factor of the pion.

Only two–pion intermediate states contribute to order q4, tadpole diagrams are not shown.q4 (to two loops), one can use dispersion relations to get the real part also at two–looporder. To arrive at the imaginary part of the normalized scalar ffto order q4, one onlyhas to keep two–pion intermediate states, i.e.Im Γπ = σ{T 00,2(1 + Re Γπ,2) + Re T 00,4}Θ(s −4m2π) + O(q6) ,(4.20)with T 00,2 and T 00,4 the ππ scattering amplitude at tree and one–loop level and Γπ,2 theone–loop expression for the scalar ff(its explicit form can be found in refs.[13,93]).

Sincethese quantities are known from the work of Gasser and Leutwyler [13], Im Γπ is fixedat order q4. The real part can then be evaluated by making use of Cauchy’s theorem,Γπ(s) = 1 + 16 < r2 >πS s + cπSs2 + s3πZ ∞4M2πds′s′3Im Γπ(s′)s′ −s −iǫ + O(q6) ,(4.21)where the scalar radius of the pion and the curvature cπS are given by< r2 >πS =38π2F 2π(¯ℓ4 −1312) +¯d1M 2π16π2F 2π+ O(M 4π)cπS =116π2F 2π19120M 2π+¯d216π2F 2π+ O(M 4π) .

(4.22)There appear two unknown parameters, these are the low energy constants ¯d1,2 relatedto the effective Lagrangian at two–loop order. They can be fixed by requiring the valuesof < r2 >πS and cπS to agree with the ones of the dispersive analysis.

This leads to¯d1 = 22 and ¯d2 = 8.9. Notice also that the coefficient ¯d2 contains infrared logarithms sothat Γπ stays finite in the chiral limit.In fig.7a,b, the real and the imaginary part of the scalar ffto two loops are shown,respectively, in comparison to the one–loop prediction and the empirical curve based on39

Fig. 7:Scalar form factor of the pion.

The curves labelled ’1’, ’2’, ’O’ and’B’ correspond to the chiral prediction to one–loop, to two–loops, the modifiedOmn`es representation and the result of the dispersive analysis, respectively.The real part is shown in (a) and the imaginary part in (b).the phase shift analysis of Au, Morgan and Pennington [94]. For the real part, the one–loop approximation reproduces Re Γπ(s) within 10 percent for energies below 400 MeVbut it fails above 450 MeV.

In contrast to the bend down of the empirical solution, itrises steadily. The two–loop curve shows the proper behaviour, the broad enhancementaround √≃550 MeV will be discussed below.At two–pion threshold, the two–loopcorrection to the one–loop enhancement is already substantial.

For the imaginary part,the one–loop result fails already very close to threshold. This is not surprising since,quite generally, the imaginary part calculated order by order in the energy expansion,only reflects the real part at one order less.

The two–loop prediction is close to theempirical result up to 560 MeV. However, since two–loop corrections are large, onemight wonder what happens with contributions of yet higher order.

I will come back tothis point. First, however, a few more comments on the real part of the scalar ff. Thereason for the turnover at √s ≃550 MeV can be understood from the fact that one canwrite Re Γπ in an Omn`es (exponential) form,Re Γπ(s) = P(s) eRe ∆0(s) cos δ00(s) + O(q6) ,(4.23)40

where the Watson final–state theorem [95] is fulfilled at next–to–leading order, Im∆0(s) = δ00(s) + O(q6). The explicit form of P(s) and ∆0(s) are given in ref.[93].

Al-though this representation is not unique, the appearance of the factor cos δ00 explains theturnover. The phase δ00, which enters at one–loop level, goes through π/2 at √s ≃680MeV forcing the real part to vanish at this energy.

This is clearly a two–loop effectsince cos δ00 = 1 −(δ00)2 + . .

. = 1 + O(s2/F 4π) and explains why the one–loop predic-tion increases monotonically.

Furthermore, in ref. [96] the physical interpretation of thebroad enhancement in Re Γπ, which is reminiscent of a resonance structure, was given.Consider the imaginary part (which does not show any resonance behaviour) and sub-tract from it the uncorrelated two–pion background.

This subtracted imaginary parthas a bell shape peaked around 600 MeV with a width of 320 MeV. The strong finalstate interactions mock up a broad, low–lying scalar–isoscalar ”particle” which is indeedneeded to furnish the intermediate–range attraction in the boson–exchange picture ofthe nucleon–nucleon interaction.

The low–lying scalar used in this type of approach istherefore nothing but a convenient representation.The exponential (Omn`es) form (4.23) suggests that higher loops in CHPT can besummed in an easy manner using the final–state theorem. However, since ∆0(s) behavesbadly at high energies and the form (4.23) is not unique, in ref.

[93] a modified Omn`esform was proposed which cures these shortcomings. It reads¯Γ(s) = ΓΛe∆Λ(s) = (1 + b · s)e∆Λ(s)∆Λ(s) = sπZ ∞4M2πds′s′Φ(s′)s′ −s −iǫtan Φ =sin 2δ00e−2η1 + cos 2δ00e−2η(4.24)where the cut offΛ is chosen around 1 GeV (below ¯KK threshold).

The reduced Omn`esfunction ∆Λ(s) takes into account the two–pion cut and therefore the reduced ffΓΛ canbe well represented by a polynomial linear in s (the slope parameter b is adjusted tothe empirical value of the scalar radius). The representation (4.24) is very accurate atlow energies.

In the chiral limit, ¯Γ(s) contains the leading and and next–to–leadingsingularities in s. In addition, at each order N in the chiral expansion, the final–statetheorem is obeyed,¯ΓN = e2iΦN ¯Γ∗N(4.25)The curves labelled ”O” in fig.7a,b show the result of the modified Omn`es representationwith Λ = 1 GeV. The real part follows the exact solution closely up to 550 MeV.

Beyondthis energy, it falls offslightly too steeply. However, would one have a more accuraterepresentation of the phase Φ, a better description of the scalar ffwould result.

Theimaginary part reproduces the exact solution within 15 per cent up to 700 MeV. Therepresentation (5.7) allows furthermore to investigate the importance of higher loops inthe following sense.

Expanding ¯Γ in powers of q2 and collecting pieces from the polynom41

and the exponential, one has ¯Γ = c0 + c2 + c4 + c6 + . .

. with c0 = 1 and c2n = O(q2n).For the real part, one finds that three loop contributions cannot be neglected beyondenergies of 500 MeV.

For the imaginary part the three loop contributions are less thanhalf of the two–loop ones (in magnitude) below 600 MeV and higher loops are negligible.This explains why the already good two–loop result is only mildly affected in this energyrange. One can perform the same study for the vector ffof the pion [93].

Below 500 MeV,the data have large error bars which makes an exact comparison difficult. However, onefinds that the two–loop result describes the data below 500 MeV.

The polynomial piecesare completely dominant, i.e. the unitarity corrections are small (in contrast to thescalar ff).

Clearly, the nearby ρ resonance starts to play a dominant role beyond thisenergy, indicating that explicit resonance degrees of freedom should be included.An attempt to incorporate the resonances in EFT was made in ref. [71] in the contextof ππ and πK scattering.

The resonance Lagrangian of Ecker et al. [32] was used and themomentum dependence of the resonance propagators was fully taken into account.

Thisleads to a good description of the data up to energies of 1 GeV and, by construction,the P–waves are in perfect agreement with the data. The phase of ǫ′ comes out tobe Φ(ǫ′) = 44.5◦, consistent with the value (4.12).

However, this calculation is notcomplete since not all effects at order q6 and higher have been taken into account (onlythe ones related to the various resonance propagators). The whole program of carryingout the loop expansion in an effective theory of pseudoscalar Goldstone bosons coupledto resonances has yet to be performed.There are some other proposals to enlarge the range of applicability of the truncatedchiral expansion.

Truong [91] has investigated the scalar and vector ffof the pion. Toenforce the final–state theorem to all orders, he considers an Omn`es–Muskhelisvhiliintegral equation for the inverse ff.This is equivalent to summing an infinite seriesof bubble graphs and formally nothing but the Pad´e approximant [0,1] (notice thatthis notation differs from the one used by Truong) of the chiral series.

This means thatF = 1+F (2)+F (4) is substituted by F[0,1] = F (2)/(1−F (4)/F (2)) with F a generic symbolfor a form factor and the subscript gives the chiral dimension. While good agreementwith the data is claimed [91], the method was criticized in ref.[93].

First, it was shownthat the Pad´e approximant sums next–to–leading order chiral logarithms with the wrongweight, i.e. it does not respect the constraints from the chiral symmetry.

Furthermore,the modified Omn`es representation discussed before gives a better description of thescalar ffthan the [0,1] approximant (and respects the next–to–leading order logs). Also,the Pad´e [0,1] apparently describes the data in the resonance region of the P–wavewell.

However, the position, width and heigth of the resonance are very sensitive to thevector radius (which is used to fix the coupling Lr9 as discussed in section 2.5). Dobado,Herrero and Truong [92] have also applied this method to unitarize the ππ scatteringamplitudes.

Dobado and Pelaez [97] have recently investigated the inverse amplitudemethod which is formally equivalent to the Pad´e [0,1] series. Their philosophy is to fixthe couplings Lr1,2 and L3 in some partial waves.

Having done that, they find a gooddescription of all phase shifts up to energies of 1 GeV in ππ and πK scattering. Notice,42

however, that the data they claim to fit well for the phase δ20(s) are outside the Royequation band discussed before. Clearly, all these unitarization methods are build onthe notion of imposing strict (elastic) unitarity on the expense of a controlled chiralexpansion.

Choosing a particular unitarization procedure induces an unwanted model–dependence and thus violates the strict requirements of CHPT. It is true, however,that higher loop calculations will definitively have to be done by unifying dispersiontheoretical methods with chiral low–energy constraints as exemplified in the calculationof the scalar ffdiscussed before.

It is always preferrable to have more loops evaluatedthan using some unitarization prescription [98].4.5. The decays K →2π and K →3πThe non–leptonic weak interactions are a wide field were CHPT methods can beapplied, in particular for the many decay modes of the kaons.I will focus here onsome recent work in connection with next–to–leading order calculations of the reactionsK →2π and K →3π (one reason being that it relates back to the ππ phase shiftsdiscussed before).

There are many other articles on kaon decays. Here, let me justmention the nice series of papers by Ecker, Pich and deRafael [99] on decays like K →πe+e−, K →πµ+µ−, K0 →π0γγ and so on.

I refer to these papers and the upcomingreview by these authors [100] for a comprehensive treatment of these and other K–decays(and the list of references therein).Let us now concentrate on the decays K →2π, 3π at next–to–leading order. Tolowest order, the Lagrangian of the strangeness-changing non–leptonic weak interactionsreadsL(2)eff(∆S = 1) = −c2Tr(λ6U †∂µUU †∂µU) −c3tjlikTr(QijU †∂µU)Tr(Qkl U †∂µU)(4.26)with (Qij)kl = δilδjk 3 × 3 matrices in flavor space, λ6 = Q32 + Q23 projects out the octetand the tjlik the 27-plet of the interactions.

This follows from the fact that the chiraltransformation properties of the weak interactions are characterized by a decompositioninto left–chiral representation as (8L, 1R)⊕(27L, 1R). The so-called weak mass term canbe neglected.

To lowest order, the constants c2 and c3 of the ∆I = 1/2 and ∆I = 3/2operators can be fixed from K →2π decays [101],c2/F 2π = 0.95 · 10−7 , c3/F 2π = −0.008 · 10−7 . (4.27)The fact that c2 >> c3 constitutes the so-called ∆I = 1/2 rule.

At next–to–leadingorder, the chiral non–leptonic Lagrangian has been worked out by Kambor, Missimerand Wyler [102]. Using chirality and CPS symmetry [103] as the basic principles, onearrives at:L(4)eff(∆S = 1) =47Xi=1Eri Q8i +33Xj=1Drj Q27j.

(4.28)43

The explicit form of the octet (Q8i ) and the 27–plet (Q27j ) operators can be found inref. [102].Not all of these counterterms are independent, but they serve as a usefulbasis.

If one switches offthe external fields, one is left with 15 (13) independent contactterms for the octet (27–plet). Clearly, a complete determination of the renormalizedweak low–energy constants Eri and Dri is not available (this is one place were the modelestimates discussed in section 2.5 are very useful).Kambor, Missimer and Wyler [104] have considered the decays K →2π and K →3π at next–to–leading order.

The appearing counterterms can be fitted from decay ratesand slope parameters. For doing that, one has to consider the isospin decomposition ofthe pertinent amplitudes.

Consider first the CP–conserving K →2π decays,A(Ks →π0π0) = ip2/3 a1/2eiδ00 −ip4/3 a3/2eiδ20A(Ks →π+π−) = −ip2/3a1/2eiδ00 −ip4/3 a3/2eiδ20A(K+ →π+π0) = −ip3/4 a3/2eiδ20(4.29)where the coefficients a1/2 and a3/2 are real. The two pions in the final state are in anS–wave with total isopsin zero or two and therefore the corresponding ππ phases appear(see the discussion after eq.(4.12)).

The five K →3π amplitudes are decomposed interms of two intercepts (α1, α3), three linear slope (β1, β3, γ3) and five quadratic slope(χ1, χ3, ξ1, ξ3, ξ′3) parameters. These twelve quantities can be expressed in terms of sevencombinations of the weak counterterms plus the strong Lr1,...,5 together with loop andtree level contributions, the latter involving the parameters c2, c3, Fπ and FK (neglectingterms of order M 2π/M 2K).

In ref. [104] the available data were refitted and then a least–square fit was used to determine the values of c2, c3 and the Ki (i = 1 .

. ., 7) from thenext–to–leading order expressions.

The most prominent result of this analysis is thatthe value for c2 is diminished by 30 per cent,c2/F 2π = 0.66 · 10−7(4.30)to one loop and the 27–plet coupling c3 remains unchanged. This reduction of c2 impliesthat a factor 1.5 in the ∆I = 1/2 enhancement is due to long distance effects (see alsorefs.[105]).

* Notice that the corresponding phase shift difference δ00 −δ20 comes out tobe 29◦, which slightly less than the tree level result of 37◦. This is an effect of the fittingprocedure since to the order the amplitudes are determined, the imaginary parts shouldcome out at their tree level values.

To recover the result (4.12) one would have to gofurther in the loop expansion. In ref.

[104] it was also pointed out that there are twodominating ∆I = 1/2 counterterms (which have previously been discussed in ref. [107])which are considerably larger than the two well–determined ∆I = 3/2 counterterms.This means that the ∆I = 1/2 enhancement is preserved at next–to–leading order, a* In ref.

[106] it was argued that no suppression of the ∆I = 3/2 piece due to long–distance effects exists.44

welcome feature. For a more detailed account of these topics, the reader should consultrefs.

[102,104].Kambor et al. [108] have further investigated these decay modes.They pointout that to order M 2π/M 2K one can derive relations which are independent of the weakcounterterms K1, .

. ., K7 at the four derivative level (besides five already known relationsat the two derivative level).

To be more precise, there are 2 (3) relations in the ∆I = 1/2(3/2) sector between various of the slope parameters. These conditions follow directlyfrom the CHPT calculation and thus are a highly non–trivial test.

The two ∆I = 1/2relations are in good agreement with the data where as the comparison in the ∆I = 3/2sector is less favorable. However, it is stressed in ref.

[108] that large electromagneticcorrections in the ∆I = 3/2 case are possible and that the analysis in this sector is moresensitive to small errors in the analysis due to cancellation of large numbers. Clearly, abetter empirical determination of the decays KL →3π, KL →π+π0π−and K+ →3πwould lead to a precision test of CHPT in the ∆I = 3/2 sector.

As a final comment,let me point out that in ∆I = 1/2 amplitudes one has large loop corrections whichare due to the strong pionic final state interactions in the isospin–zero S–wave. It isconceivable that a resummation technique as discussed in section 4.4 will allow to sumthese rescatterig diagrams in harmony with chiral symmetry.5.

FINITE TEMPERATURES AND SIZESCHPT allows to make precise statements about the low temperature behaviour ofthe strong interactions. Furthermore, Goldstone boson induced finite size effects can becalculated in a controlled fashion which is of importance for lattice gauge calculations.In this section, I will first discuss some aspects of finite temperatures, mostly the meltingof the quark and gluon condensates with increasing temperature.

Then, I will cover somedevelopments concerning finite size effects, in particular the comparison of recent MonteCarlo studies of σ models and the chiral predictions. Technical details will generally beomitted and the reader should consult the pertinent references.5.1.

Effective theory at finite temperatureIn the chiral limit, the pions are massless and dominate the spectrum. At suffi-ciently low temperatures, they do not interact forming a Bose gas with the pressuredirectly proportional to the fourth power of the temperature.

Interactions among thepions generate power-like corrections which are controlled by the parameter T 2/8F 2πand the contributions of the heavy particles are exponentially suppressed (see below fordetails). If the temperature is sufficiently small, a perturbative analysis of these effectscan be performed making use of CHPT techniques as developed by Gerber, Gasser andLeutwyler [109,110,111] (see these papers for references on earlier work on this subject).Instead of the S–matrix, one deals with the generating functional Z defined viaZ = Tr [e−H/T ](5.1)45

with T the temperature and H the Hamiltonian. The quantity in the square bracketsis nothing but the analytic continuation of the time evolution operator to the pointt = −i/T.

This allows to write(U2|e−tH|U1) =Z[dU] exp−Zd4x ˜Leff(5.2)with U1 = U(⃗x, 0), U2 = U(⃗x, t) and ˜Leffcan be obtained from Leffby replacing theMinkowski metric gµν by −δµν and changing the overall sign. To perform the trace,one sets U1 = U2 and integrates over U1 over an interval of length 1/T.

The functionalintegral then extends over all field configurations which are periodic in the time direction,Z =Z[dU] exp{−ZMd4x ˜Leff}U(⃗x, x4) = U(⃗x, x4 + β)(5.3)with β = 1/T. The pions live on the manifold M which is nothing but the torus R3×S1.The circumference of S1 is given by the inverse temperature β.

It is important to realizethat the coupling constants like F, B, Lr1, Lr2, . .

. remain unaffected (are temperatureindependent).

The boundary conditions in the direction of x4 are dictated by the tracethat defines the partition function (strictly speaking, the trace operation only makessense at finite volume. The finite size effects generated by the box are discussed inref.[112]).

The only change induced by the temperature is a modification of the pionpropagator. In the chiral limit and in euclidean space, the T–dependent propagator isgiven byG(x) =∞Xn=−∞∆(⃗x, t + nT )(5.4)where ∆(x) = [4π2(⃗x2 + t2)]−1 is the T = 0 propagator.

Any Green function can beevaluated at finite T along the same lines. Denote by A an arbitrary operator (like e.g.the product of two currents), its thermal expectation value is simply given by< A >T = Tr[e−βH A]Tr[e−βH](5.5)In particular, the free energy density z readsz = ǫ0 −P = −TlimV →∞ln ZV(5.6)46

Fig. 8:Typical two and three loop diagrams for calculating the free energydensity.

The black dots (•) denote insertions from L(2)eff.with ǫ0 the energy density of the vacuum and P the pressure. Clearly, to evaluate thepartition function to some given order in the temperature, one has to evaluate CHPT tothe corresponding order in the loop expansion.

To make this statement more transpar-ent, consider some typical Feynman diagrams contributing to z (fig.8). They have noexternal legs, however, instead of the small external momenta one now has to deal withtemperature insertions.

The following remarkable feature emerges [111,113]. Tree dia-grams from L(2,4,6,...)effare temperature independent and therefore only contribute to thevacuum energy.

If one now evaluates the free energy density to order qn, countertermsfrom L(n−2)effenter through one–loop graphs and thus give rise to a T–dependent contri-bution. This can, however, be absorbed in the renormalization of the pion mass [111].Therefore, after subtracting the vacuum energy density and expressing the remainingcontributions in terms of the physical pion mass, the free energy density evaluated toorder qn only involves low–energy constants from the effective Lagrangian up to andincluding order n −4.

This makes a three loop calculation feasible since only the knownone–loop couplings enter [111]. The calculations are performed in dimensional regu-larization.In that case, the temperature independent part of the pion propagagtorgenerates all the singularities as d →4 (see ref.

[114] for details). Furthermore, mostthermodynamic quantities can be expressed in terms of powers of the thermal propaga-tor at x = 0 and the functions gr associated to the noninteracting d–dimensional Bosegas,gr(M, T) = 2Z ∞0dλ(4πλ)d/2 λr−1 exp(−λM 2)∞Xn=1exp(−n2/4λT 2) .

(5.7)This essentially specifies all the tools one needs for finite temperature calculations (fora more detailed account, see ref.[111]).5.2. Melting condensatesThe quark condensate is the thermodynamic variable conjugate to the mass of thelight quarks.

It is given by< ¯qq >T = Tr [¯qq exp(−H/T)]Tr [exp(−H/T)]=∂z∂mq(5.8)47

where mq denotes the quark mass. The quark condensate plays the same role as themagnetization in a ferromagnet.Following this analogy, it is supposed to decreaseas temperature increases and eventually to disappear.Disorder takes over and thesymmetry is restored.

In the case of QCD at low temperatures, the pions dominate theproperties of the system since the contributions of the massive states to the partitionfunction are exponentially suppressed. The free energy density admits an expansion inpowers of T 2 and log T much like the ππ scattering amplitude in powers of the externalmomenta and logs thereof:z =Xm,ncm,n(T 2)m (T 2 logT)n + O(e−¯M/T )(5.9)with m, n = 0, 1, 2, .

. .

and ¯M denotes the mass of the lightest massive state (like thekaon if one works in the two–flavor case). Clearly, eq.

(5.9) is an asymptotic series, theexponentially suppressed contributions from the massive states can never be accountedfor by powers of T 2 or logarithms. This is completely analogous to perturbative QCDseries, where instantons generate non–perturbative corrections of the type exp(−8π/g2).From eq.

(5.8) it follows immediately that the thermal expectation value of the quarkcondensate has a similar representation.For Nf massless quarks, the expansion of< ¯qq >T reads< ¯qq >T =< 0|¯qq|0 >1 −c1 T 28F 2π−c2 T 28F 2π2 −c3 T 28F 2π3 ln(ΛqT ) + O(T 8)(5.10)withc1 = 23N 2f −1Nf, c2 = 29N 2f −1N 2f, c3 = 827(N 2f + 1) Nf . (5.11)The coefficients c1, c2 and c3 were first worked out by Bin´etruy and Gaillard [115],Gasser and Leutwyler [109] and Gerber and Leutwyler [111], respectively.

For masslessquarks, this result is exact. The first two terms in the expansion are entirely given interms of the parameter Fπ which characterizes the lowest order effective Lagrangian.Notice also the appearance of the characteristic temperature Tc =√8Fπ ≃250 MeV.Only at next–to–next–to–leading order a logarithm appears.

The unknown scale canbe related to the SU(2) low energy constants ¯ℓ1,2 of the next–to–leading order effectiveLagrangian. These couplings are related to the isospin–zero D–wave scattering lengtha02 viaa02 =1144π3F 4πln( ΛqMπ) −0.067 + O(M)(5.12)which leads to Λq = 470 ± 110 MeV [113].

In figure 9, the one, two and three loopresults are shown for temperatures below 150 MeV (using the central value of Λq). Athigher temperatures, the corrections to the tree result become so large that the ex-pansion (5.10) does not make sense any more.

In particular, CHPT can not predict48

Fig. 9:The quark condensate at finite temperature (after ref.[111]).

The solid,dashed and dashed–dotted line represent the one, two and three loop calcula-tions. The error bar at the three loop curve at 150 MeV represents the uncer-tainty in the scale of the chiral logarithm.the temperature of the chiral phase transition, at that point the corrections eat up theleading term completely.

The case of non–zero quark masses has also been discussedby Gerber and Leutwyler [111]. As expected, a finite pion mass slows down the melt-ing of the quark condensate.

Even though the quark masses are tiny, the temperaturedependence is strongly affected because the corresponding Boltzmann factor involvesMπ rather than M 2π ∼ˆm. Furthermore, these authors also discuss the influence of themassive states (K, η, ρ, N, .

. .) making use of a dilute gas approximation.

Below T = 150MeV, the massive states influence the melting of the condensate very little because ofthe Boltzmann factors. Beyond this temperature, one can not neglect the interactionsbetween the massive states and the ones to the pions any more.

To summarize, chiralsymmetry predicts the temperature dependence of the quark condensate. As the tem-perature increases, it gradually melts.

At a temperature of approximately 150 MeV, theloop corrections (in the three loop approximation) have decreased the condensate abouta factor of two, rendering the perturbative analysis useless beyond this point. However,if one ignores that for a moment and follows the result to higher temperatures, thecondensate vanishes at a temperature of 170 .

. .190 MeV.In a similar fashion, one can study the temperature dependence of the gluon con-densate [116,117] (which is expected to be much weaker than the one of the quarkcondensate as indicated by some models [118]).

For that, one considers the trace of theenergy–momentum tensor,Θµµ = −β(g)2g3 GaµνGµν,a + {1 + γ(g)} ¯qMq + c 1(5.13)49

Here, β(g) is the QCD β–function, µdg/dµ = β(g). The first term in (5.13) accountsfor the fact that the strong coupling constant is scale dependent (conformal anomaly)and the second one is due to the explicit scale breaking from the quark masses.

The lastterm is fixed by the normalization condition that the trace of the energy–momentumtensor is zero in the vacuum, i.e. c =< 0|G2|0 >, with G2 = −βGaµνGµν,a/2g3.

At finitetemperature, one has< G2 >T =< 0|G2|0 > −< Θµµ >T(5.14)Two comments are in order. First, the gluon condensate is not an order parameter,conformal symmetry is also broken in the high temperature phase of QCD.

Second, thevev < 0|G2|0 > can not be determined very accurately, but its value drops out if oneconsiders the temperature dependence of the gluon condensate. The quest is now tocalculate < Θµµ >T .

One can make use of the thermodynamic relations s = dP/dT andǫ = Ts −P, with s the entropy density. Therefore, the knowledge of the equation ofstate P = P(T) is all one needs [111,112,116],< Θµµ >T = ǫ −3P = T 5 ddT PT 4(5.15)Since the correction to the free Bose gas for the massless Goldstone bosons starts out atorder T 8, one finds a weak temperature dependence of the gluon condenstae.

For twomassless quark flavors, one has [116]< Θµµ >T = π2270T 8F 4πlnΛpT−14+ . .

. (5.16)where the ellipsis stands for higher terms in the temperature expansion and the contri-bution from the exponentially suppressed massive states.

One notices that the gluoncondensate indeed admits a very weak temperature dependence. The physical reason isthat the operators Θµµ and G2 are chiral singlets and thus single pion emision/absorptionis not allowed.

Furthermore, < Θµµ >T vanishes up to and including two loops in CHPT.This leads to the leading T 8 term in (5.16). Clearly, since the pion contribution is somuch suppressed, the massive states play a more important role already at lower tem-peratures.

For a detailed discussion of these topics, see Gerber and Leutwyler [117].Finally, one can also study the kinetics of the hot pion gas and its relation to theassumed quark/gluon plasma transition. This topic will not be addressed here, I referto the article by Goity [119] which also includes references to pertinent work in thatfield.5.3.

Effective theory in a box50

One can also address the following question: What happens if the pion fields areconfined in a box of finite volume? At first sight, this might sound academical since oneknows that there is no spontaneous symmetry breaking in a finite volume.

However,CHPT can be used to understand how the infinite volume limit is approached. Thisis of particular relevance for lattice simulations.

On the lattice, the world is a four-dimensional box.In general, one has to deal with finite size effects induced by theparticles of mass M which are the lowest excitations in the spectrum. A general analysisof these finite size effects and the approach to the continuum limit has been performedby L¨uscher [120] for the case ML >> 1 (L denotes the size of the box).

Matters are,however, different in the presence of Goldstone bosons (massless excitations). No matterhow large one choses the lattice size, the parameter ML will always be small and finitesize effects will be large.

This can be most easily understood in the case of the quarkcondensate. For fixed quark masses and zero temperature, the vev < ¯qq > is always zeroas long as L is finite.

Only when L →∞, symmetry breaking occurs and < ¯qq ≯= 0.Obviously, the finite size effect < ¯qq >L −< ¯qq >∞is as big as the quantity itself (seealso the discussion by Leutwyler in ref.[116]). As will be shown later on, CHPT canbe used to systematically calculate the Goldstone boson induced finite size effects in anexpansion in powers of 1/L2 (in four dimensions).

Ultimately, this will become a tool tocontrol finite size effects in the study of lattice QCD with reasonably light quarks. Atpresent, these methods are tested versus Monte Carlo data on O(N) σ models as willbe discussed below.The effective Lagrangian at finite volume V = L1 ×L2 ×L3 ×L4 = L1 ×L2 ×L3 ×βhas been studied in detail by Gasser and Leutwyler [109,112] and also by Neuberger[121].

The lowest order term in the effective theory which generates the leading lowenergy contributions to the Green functions has the standard form involving the twocoupling constants F and B. The pion field is subject to periodic boundary conditions,U(x) = U(x + n) , n = (n1L1, n2L2, n3L3, n4β)(5.17)with the ni integer numbers.

The constants F and B are not affected by the finitevolume [112]. As before, the only modification appears in the pion propagator, it takesthe formG(x) =XnG0(x + n)(5.18)with G0(x) the free (T = 0, L = ∞) propagator.

The quantities Mπ, T and 1/Li are allbooked as O(p) at fixed ratios Mπ/T and LiT (the so–called ”p–expansion”). Effectively,one seeks an expansion of the generating functional in powers of these small quantities(a complication arises due to zero modes as discussed below).

At next–to–leading order,matters are more complicated. Since the box breaks Lorentz invariance, one expects ad-ditional terms related to this phenomenon and also boundary terms which are sensitiveto the boundary conditions on the fields.

There exists, however, a tremendous simpli-fication if one choses a rectangular box and imposes the same boundary conditions on51

the fields on the walls of the box as in the time direction proportional to L4. Then, thepartition function is invariant under permutations of the walls of the four–dimensionalbox.

Since we had already seen that the low–energy constants do not depend on thetemperature, one can use this symmetry to show that they also do not depend on thefinite volume. Furthermore, no Lorentz non–invariant vertices and surface terms appearin this case.

Therefore, the effective Lagrangian is just the one used at zero temperaturein the infinite volume limit.Let us now consider Nf quarks of the same mass and denote the associated Gold-stone boson mass by M. The partition function takes the form (for L1 = L2 = L3 =L)[109]Tr e−βH = e−βL3zz = ǫ0 −N 2f −12g0(M 2, T, L) +N 2f −14NfM 2F 2 [g1(M 2, T, L)]2 + O(p8)(5.19)with ǫ0 the energy density of the groundstate. The functions g0 and g1 are defined viagr(M 2, T, L) =116π2Z ∞0dλλr−3 Xn̸=0exp (−M 2λ −n2/4λ)(5.20)These take the familiar form known from the relativistic Bose gas in the infinite volumelimit.

It is important to omit the term with n1 = n2 = n3 = n4 = 0 from the sumin (5.20). In the chiral limit, the system develops zero modes and consequently thefunction g1 exhibits a pole,g1(M 2, T, L) =1M 2L3β + ¯g1(M 2, T, L)(5.21)where ¯g1 is finite when M vanishes.

This pole is due to the propagation of the zeromodes. Their contribution to the functional integral is not of the Gaussian type as canbe seen fromZd4xL(2) = −12NfF 2M 2V + 14F 2Zd4x Tr (∂µφ∂µφ + M 2φ2) + O(p4)(5.22)The zero modes can therefore not be treated perturbatively.

To deal with them, oneintroduces collective coordinates and reorders the chiral expansion by enhancing thezero–mode propagators (”ǫ–expansion”)T = O(ǫ) , 1/L = O(ǫ) , M = O(ǫ2)(5.23)Expanding now the partition function in powers of ǫ, the graphs involving exclusivelyzero modes are of order one whereas graphs with non–zero modes in the propagators52

are suppressed by at least ǫ2 with the exception of the one–loop graphs which enter inthe normalization A of the final result [109]Tr e−βH = A XNf(s) , s = 12F 2M 2V(5.24)with XNf(s) the integral over the flavour groupXNf(s) =ZSU(Nf )dµ(U) exp (sRe Tr U)(5.25)and dµ(U) is the Haar measure. One arrives at this result by the standard procedureintroducing collective coordinates, U = ueiξ(x)u.

Here, ξ(x) collects the non–zero modesand the zero modes are described by the constant matrix u ∈SU(Nf). For details ofthis procedure, see Gasser and Leutwyler [109], Hasenfratz and Leutwyler [112] or theoriginal work by Polyakov [123].

In the framework of the ǫ–expansion, one can dis-cuss the symmetry restoration when the quark masses are much smaller than 1/volume[109,112,116]. With this, we have all tools at hand to calculate finite size effects.

Inthe following section, we will compare some theoretical predictions with the results ofMonte Carlo studies.5.4. Finite size effects: CHPT versus Monte CarloHere I will briefly discuss the comparison of theoretical predictions of Goldstoneboson induced finite size effects with recent numerical simulations.

For all technicaldetails, I refer the reader to the pertinent articles. First, consider models with brokenO(N) symmetries in more than two dimensions.

These are of relevance in the studyof the Higgs mass bounds, two–flavour QCD or finite size effects in ferromagnets closeto the critical point. Hasenfratz and Leutwyler [112] have given theoretical predictionsfor the free energy, magnetization, susceptibilities and two–point functions up to andincluding order (1/Ld−2)2 corrections for d = 3 and d = 4.In the first case, onlytwo low–energy constants enter whereas in four dimensions, one has to deal with threeadditional ones.

They use the linear σ–model in d dimensions making use of ”magnetic”language,L = 12∂µφa∂µφa + 12m2φaφa + 14λ(φaφa)2 −Hφ0(5.26)The real field φ has N (a = 0, . .

., N −1) components and the external magentic fieldH = (H, 0, . .

., 0) breaks the O(N) symmetry explicitely.As H tends to zero, weare interested in the coupling constant regime where spontaneous symmetry breakingO(N) →O(N −1) occurs signaled by a non–vanishing value of the magnetisation Σ,limH→0 < φ0 >= Σ , < φi >= 0(5.27)53

Translated to QCD language, Σ is the vev of the scalar quark density and H is equivalentto the quark mass matrix. The corresponding effective theory is formulated in terms ofa vector field Sa(x) which embodies the N −1 Goldstone fields and is subject to theconstraint Sa(x)Sa(x) = 1.

The low energy properties of this model and the finite sizeeffects at next–to–leading order are discussed in detail in ref.[122]. In three dimension,there are two low energy constants entering and in d = 4 there are five, the extra threebeing related to scales of logarithmic corrections.For the d = 3 classical O(3) Heisenberg model in the broken phase near the criticalpoint, some of these predictions were confronted with Monte Carlo data by Dimitrovicet al.

[124]. They found that the finite size behaviour of the magnetisation and thecorrelation functions are in good agreemnet with the CHPT predictions.

Furthermore,they could determine the critical indices for the correlation length and magnetisation(in agreement to previous studies using other methods). In the case of the O(4) modelin four dimensions, numerical studies have been performed by Hasenfratz et al.

[125].Good agreement with the theoretical predictions is found and it is shown that in situ-ations where the Goldstone bosons control the dynamics of the system, one can indeeddetermine the infinite volume, zero external source quantities from the finite volume sim-ulations in a controlled way. Another quantity of interest is the so–called constrainedeffective potential which in the context of the non–linear σ–model has been studied byG¨ockeler and Leutwyler [126].

A numerical simulation has been presented by Dimitrovicet al. [127].

They found good agreement for the shape of the constrained effective po-tential in the vicinity of its minimum, but also noted that this method is not as accuratein determining the low energy constants Σ and F than the one presented in [125].Hansen [128] has extended this analysis to theories with SU(N) × SU(N) brokensymmetries and Hansen and Leutwyler [129] have studied the charge correlations andtopological susceptibility of QCD at finite volume and temperature. It is conceivable thatthese predictions will become useful when one will be able to perform QCD simulationswith dynamical fermions of reasonably small masses.

At present, many Monte Carlostudies are done in the quenched approximation (i.e. suppressing fermion loops).

In thatcase, the theoretical predictions are not applicable. Effective theories in the quenchedapproximation are being developed by Sharpe [130] and Bernard and Golterman [131].Finally, Leutwyler and Smilga [132] have discussed the spectrum of the Dirac operatorand the role of the winding number in QCD.

The distribution of winding number and thespectrum of the Dirac operator at small eigenvalues are related to the quark condensateat infinite volume, which implies that the formation of the quark condensate is connectedto the occurrence of eigenvalues of the order λn ∼1/V . What their analysis can notprovide is a reason why a condensate arises.

For a more detailed discussion of these andother topics, the reader should consult ref.[132].5.5. An application to high–Tc superconductivity54

To demonstrate the universality of the CHPT methods, let me briefly discusssome work by Hasenfratz and Niedermayer [133] on the correlation length of the anti–ferromagnetic (AF) d = 2 + 1 Heisenberg model at low temperatures.One of theirmotivations is the recent discovery of quasi–two–dimensional anti–ferromagnets like theundoped La2CuO4 compound. In this crystal, a single unpaired electron at each Cu++site makes the material antiferromagnetic with a quasi–two–dimensional structure.

Thismeans that the forces in the plane are much stronger than the forces between neigh-bouring planes. As it is well known, in doped La2CuO4 one observe high Tc super-conductivity.

Doping e.g. with Sr atoms renders the planes metallic and leads to thesuperconductivity.To understand this phenomenon one has to understand first thephysics of the undoped La2CuO4 system.

Due to the particular structure of this crys-tal, one can use CHPT methods to understand its low temperature behaviour. It shouldbe well described by the d = ds + 1 = 2 + 1 dimensional quantum Heisenberg model,H = JX⃗Si⃗Sj (J > 0)(5.28)where ⃗Si is the spin operator at site i and ⃗S2i = S(S + 1).

In the case of La2CuO4, onehas S = 1/2. The groundstate is assumed to be antiferromagnetically ordered (whichis strictly proven only for |S| ≥1).

In the case of a non–vanishing value of the cou-pling strength J, the rotational symmetry is broken and one has two (almost) masslessGoldstone bosons (magnons). These modes pick up a small mass since the Mermin-Wagner-Coleman theorem [134] forbids massless Goldstone bosons in two dimensionsat finite temperature.

These magnons interact weakly at low temperatures, T << J.This makes an analytic study feasible. Note, however, that at very low energies (muchsmaller than T) the magnon interaction starts to become stronger again which leads toa finite mass gap in the theory.

The exact result for the mass gap of the d = 2 O(3)non-linear σ–model has been given in ref. [135] and can also be used in the present case.The temperature dependence of the correlation length ξ can now be calculated from aneffective Lagrangian which shares the properties of the quantum AF model (5.28).

Thedetails are given in ref. [133] and the final result reads:ξ(T) = e8¯hc2πF 2 exp(2πF 2/T)1 −12T2πF 2 + O(T2πF 2 )2(5.29)with F 2 the spin–stiffness and c the spin–velocity (notice that in d = 3, F has dimension[energy]1/2).

The result (5.29) applies for the temperature range TN < T << 2πF 2, withTN ≃300 K the temperature until which the La2CuO4 compound remains in the brokenphase. A recent measurement of the spin–velocity gives ¯hc = (850 ± 30)meV ˚A [136].Notice that the temperature dependence of ξ(T) is essentially given by the exponentialprefactor (which was already predicted in ref.

[137]), at 600 K the correction T/4πF 2is ∼17 percent. The important prefactor e/8 comes from the exact form of the mass55

Fig. 10:Inverse correlation length of undoped La2CuO4.

The solid line isthe CHPT prediction, eq. (5.29), and the dashed one accounts only for the firstterm in the square brackets.

The data are taken from refs.[138,139].gap. In fig.10, a one–parameter fit to the recent data is shown, one finds for the spin–stiffness 2πF 2 = 150 meV = 1740˚A [138].Clearly, the fit is not perfect, but thiscan not be expected.Anisotropies in the spin–spin interactions might lead to suchdeviations.Phenomenologically, it was found that for doped samples the followingrule holds: ξ−1(x, T) = ξ−1(x, 0) + ξ−1(0, T) with x the doping concentration and theformula (5.29) applies to x = 0.

For a further discussions, see refs. [137] and the talkby Hasenfratz [139].Wiese and Ying [140] have recently performed a Monte Carlosimulation of the 2-d antiferromagnetic quantum Heisenberg model.

They compare thefinite size and temperature effects with the ones given by the chiral perturbation theorycalculation of Hasenfratz and Niedermayer [140]. They find ¯hc = (845 ± 32) meV˚A and2πF 2 = (157 ± 7) meV, in good agreement with the data and the one–parameter fitdescribed above.

They also show results for the ground state energy density and thestaggered magnetisation. This again demonstrates the usefulness of the EFT approach.56

VI. BARYONS6.1.

Relativistic formalismIn this section, I will be concerned with the inclusion of baryons in the effectivefield theory. The relativistic formalism dates back in the early days, see e.g.

Weinberg[10], Callan et al. [11], Langacker and Pagels [14] and others (for reviews, see Pagels[8] and the book by Adler and Dashen [4]).

The connection to QCD Green functionswas performed in a systematic fashion by Gasser, Sainio and ˇSvarc [142] (from hereon referred as GSS) and Krause [143]. As done in the GSS paper, I will outline theformalism in the two–flavor case, i.e.

for the pion–nucleon (πN) system. The extensionto full flavor SU(3) is spelled out in appendix B.There is a variety of ways to describe the transformation properties of the spin–1/2baryons under chiral SU(2) × SU(2).

All of them lead to the same physics. However,there is one most convenient choice (this is discussed in detail in Georgi’s book [21]).Combine the proton (p) and the neutron (n) fields in an isospinor ΨΨ =pn(6.1)The Goldstone bosons are collected in the matrix–valued field U(x).

In the previoussections, we had already seen that the self–interactions of the pions are of derivativenature, i.e. they vanish at zero momentum transfer.

This is a feature we also want tokeep for the pion–baryon interaction. It calls for a non–linear realization of the chiralsymmetry.Following Weinberg [10] and Callan et al.

[11], we introduce a matrix–valued function K. It not only depends on the group elements L, R ∈SU(2)L,R, butalso on the pion field (parametrized in terms of U(x)) in a highly non–linear fashion,K = K(L, R, U). Since U(x) depends on the space–time coordinate x, K implicitelydepends on x and therefore the transformations related to K are local.

To be morespecific, K is defined viaLu = u′K(6.2)with u2(x) = U(x) and U ′(x) = RU(x)L† = u′2(x). The transformation properties ofthe pion field induce a well–defined transformation of u(x) under SU(2) × SU(2).

Thisdefines K as a non–linear function of L, R and π(x). K is a realization of SU(2)×SU(2),K =√LU †R†R√U(6.3)and the baryon field transforms asΨ →K(L, R, U)Ψ(6.4)The somewhat messy object K can be understood most easily in terms of infinites-imal transformations.For K = exp(iγaπa), L = exp(iαaπa) exp(iβaπa) and R =exp(−iαaπa) exp(iβaπa) (with γa, αa, βa real) one finds,⃗γ = ⃗β + i[⃗α,⃗π]/Fπ + O(⃗α2, ⃗β2,⃗π2)(6.5)57

which means that the nucleon field is multiplied with a function of the pion field. Thisgives some credit to the notion that chiral transformations are related to the absorptionor emission of pions.To construct now the lowest order effective Lagrangian of thepion–nucleon system, one has to assemble all building blocks.

These are the mesonsand baryon fields, U(x) and Ψ(x), respectively as well as the appropriate covariantderivatives (I restrict myself here to the case with only scalar and vector external fields.The more general expressions are given in Appendix B),∇µU = ∂µU −ieAµ[Q, U]DµΨ = ∂µΨ + ΓµΨΓµ = 12u†(∂µ −ieAµQ)u + u(∂µ −ieAµQ)u†(6.6)with Q = diag(1, 0) the nucleon charge matrix and vµ = e(1 + τ3)Aµ/2 the externalvector field (Aµ denotes the photon field). Dµ transforms homogeneously under chiraltransformations, D′µ = KDµK†.

The object Γµ is the so–called chiral connection. It isa gauge field for the local transformationsΓ′µ = KΓµK† + K∂µK†(6.7)The connection Γµ contains one derivative.

One can also form an object of axial–vectortype with one derivative,12uµ = i2(u†∇µu −u∇µu†) = i2{u†, ∇µu} = iu†∇µUu†(6.8)which transforms homogeneously, u′µ = KuµK†. The covariant derivative Dµ and theaxial–vector object uµ are the basic building blocks for the lowest order effective theory.Before writing it down, let us take a look at its most general form.

It can be written asa string of terms with an even number of external nucleons, next = 0, 2, 4, . .

.. The termwith next = 0 obviously corresponds to the meson Lagrangian (2.9) so thatLeff[π, Ψ, ¯Ψ] = Lππ + L ¯ΨΨ + L ¯ΨΨ ¯ΨΨ + .

. .

(6.9)Typical processes related to these terms are pion–pion, pion–nucleon and nucleon–nucleon scattering, in order. In what follows, we will mostly be concerned with processeswith two external nucleons,L ¯ΨΨ = LπN = ¯Ψ(x)D(x)Ψ(x)(6.10)The differential operator D(x) is now subject to a chiral expansion.

Its explicit formto lowest order follows simply by combining the connection Γµ and the axial–vector uµ58

(which are the objects with the least number of derivatives) with the appropriate baryonbilinearsL(1)πN = ¯ΨD(1)Ψ= ¯Ψ(iγµDµ −◦m + i2◦gAγµγ5uµ)Ψ(6.11)The effective Lagrangian (6.11) contains two new parameters. These are the baryonmass◦m and the axial–vector coupling ◦gA in the chiral limit,m = ◦m[1 + O( ˆm)]gA = ◦gA[1 + O( ˆm)](6.12)Here, m = 939 MeV denotes the physical nucleon mass and gA the axial–vector strengthmeasured in neutron β–decay, n →pe+¯νe, gA ≃1.26.

The fact that◦m does not vanishin the chiral limit (or is not small on the typical scale Λ ≃Mρ) will be discussedbelow. Furthermore, the actual value of◦m, which has been subject to much recentdebate, will be discussed in the context of pion–nucleon scattering.

The occurence ofthe constant ◦gA is all but surprising. Whereas the vectorial (flavor) SU(2) is protectedat zero momentum transfer, the axial current is, of course, renormalized.

Together withthe Lagrangian L(2)ππ (2.9), our lowest order pion–nucleon Lagrangian reads:L1 = L(1)πN + L(2)ππ(6.13)To understand the low–energy dimension of L(1)πN, we have to extend the chiral countingrules of section 2.3 to the various operators and bilinears involving the baryon fields.These are:◦m = O(1) , Ψ, ¯Ψ = O(1) , DµΨ = O(1)¯Ψγµγ5Ψ = O(1) , ¯ΨσµνΨ = O(1) , ¯Ψσµνγ5Ψ = O(1)(i̸D −◦m)Ψ = O(p) , ¯Ψγ5Ψ = O(p)(6.14)Here, p denotes a generic nucleon three–momentum. Since◦m is of order one, baryonfour–momenta can never be small on the typical chiral scale.

Stated differently, anytime derivative D0 acting on the spin–1/2 fields brings down a factor◦m. However, theoperator (i̸D −◦m)Ψ counts as order O(p).

The proof of this can be found in ref. [143]or in the lectures [144].To exhibit the physics content of the Lagrangian L(1)πN (6.11), let me parametrizethe pions in the “σ–model” gauge (which is a convenient choice),U = (σ + iπ)/F , σ2 + π2 = F 2(6.15)and expand the connection Γµ as well as the axial–vector uµ in powers of the pion fields:iΓµ = vµ +i8F 2 [π , ∂µπ] −18F 2 [π , [π , vµ]] + .

. .12uµ = −12F ∂µπ +i2F [vµ , π] −116F 3 {π , {π , ∂µπ}} + .

. .

(6.16)59

Fig. 11:Feynman rules in the σ–model gauge.

Nucleons, pions and photonsare denoted by solid, dashed and wiggly lines, respectively.60

The corresponding vertices are shown in fig.11. Clearly, one recovers some well–known vertices like the pseudovector (derivative) πN coupling or the Kroll–Rudermanterm which plays an important role in pion photoproduction.

Notice that all prefactorsare given in terms of the four lowest–order constants, F, B, ◦gA and ◦m. To lowest order,the Goldberger–Treiman relation [145] is exact, ◦gA◦m = ◦gπNF, which allows us to writethe πN coupling in the more familiar form ∼◦gπN∂µπa.It goes without saying that we have to include pion loops, associated with L1 givenin (6.13).

The corresponding generating functional reads [142]exp{iZ} = NZ[dU] expiZdxL(2)ππ + iZdx¯ηS(1)ηD(1)acS(1)cb = δabδ(4)(x −y)(6.17)with η, ¯η external Grassmannian sources and S(1) the inverse nucleon propagator relatedto D(1) (6.11). a, b and c are isospin indices.

This generating functional can now betreated by standard methods. The details are spelled out by GSS [142].

I will nowconcentrate on the low–energy structure of the effective theory which emerges. Pionloops generate divergences, so one has to add counterterms.

This amounts toL1 →L1 + L2L2 = ∆L(0)πN + ∆L(1)πN + L(2)πN + L(3)πN + L(4)ππ(6.18)Let me first discuss the last three terms on the r.h.s.of eq. (6.18).These are theexpected ones.The structure of the πN interaction allows for odd powers in p, sostarting from L(1)πN to one–loop order one expects couterterms of dimension p2 and p3.A systematic analysis of all these terms has been given by Krause [143].

The first twoterms, ∆L(0)πN and ∆L(1)πN, are due to the fact that the lowest order coefficients ◦m and ◦gAare renormalized (by an infinite amount) when loops are considered. This is completelydifferent from the meson sector, the constants B and F are not renormalized by theloops.

The origin of this complication lies in the fact that the nucleon mass does notvanish in the chiral limit. To avoid any shift in the values of◦m and ◦gA one thus has toadd∆L(0)πN = ∆◦m ◦mF2¯ΨΨ , ∆L(1)πN = ∆◦gA ◦mF2 12¯Ψiγµγ5uµΨ(6.19)The first term in (6.19) can be easily worked out when one considers the nucleonself–energy ΣN(p) related to the nucleon propagator via S(p) = [ ◦m−̸p−ΣN(p)]−1 in theone–loop approximation [142].

The low–energy structure of the theory in the presenceof baryons is much more complicated than in the meson sector. This becomes mosttransparent when one compares the ππ and πN scattering amplitudes, Tππ and TπN,respectively.

While T treeππ∼p2 and T n−loopππ∼(p2)n+1, the corresponding behaviour forTπN is shown in fig.12 [142]. Here, p denotes either a small meson four–momentum or61

Fig. 12:Chiral expansion for the πN scattering amplitude, TπN ∼pN.

Treegraphs contribute at N = 1, 2, 3 . .

., n–loop graphs at N= 2,3, . .

. (aftermass and coupling constant renormalization).

The contributions from 2,3,. . .loops are analytic in the external momenta at order p3 (here, p is a pion four-,nucleon three–momnentum or the pion mass).mass or a nucleon three–momentum.

Tree graphs for TπN start out at order p followedby a string of higher order corrections p2, p3, . .

.. One–loop graphs start out at order p2(after appropriate mass and coupling constant renormalization) and are non–analyticin the external momenta at order p3 (in the chiral limit ˆm = 0). Higher loops startout at p2 and are analytic to orders O(p2 , p3).

This again means that the low–energyconstants associated to L(2,3)πNwill get renormalized.Evaluation of one–loop graphsassociated with L1 therefore produces all non–analytic terms in the external momentaof order p3 like e.g. leading threshold or branch point singularities.

Let us now considerthe case ˆm ̸= 0.Obviously, the πN amplitude also contains terms which are non–analytic in the quark masses. A good example is the Adler–Weisberger relation in itsdifferential form – it contains a factor F −2πand therefore a term which goes like ˆm ln ˆm.Due to the complicated low–energy structure of the meson–baryon system, it has neverbeen strictly proven that one–loop graphs generate all leading infrared singularities, inparticular the ones in the quark masses.

However, in all calculations performed so far theopposite has never been observed. In any case, the exact one–to–one correspondencebetween the loop and small momentum expansion is not valid in the meson–baryonsystem if one treats the baryons fully relativistically.

This can be overcome, as willbe discussed in the next section, in an extreme non–relativistic limit. Here, however, Iwish to point out that the relativistic formalism has its own advantages.

Two of themare the direct relation to dispersion theory and the inclusion of the proper relativistickinematics in certain processes. These topics will be discussed later on.

Let me make afew remarks concerning the structure of the nucleons (baryons) at low energies. Starting62

from a structureless Dirac field, the nucleon is surrounded by a cloud of pions whichgenerate e.g. its anomalous magnetic moment (notice that the lowest order effectiveLagrangian (6.11) does only contain the coupling of the photon to the charge).

Besidesthe pion loops, there are also counterterms which encode the traces of meson and baryonexcitations contributing to certain properties of the nucleon. Finally, one point whichshould be very clear by now: One can only make a firm statement in any calculationif one takes into account all terms at a given order.

For a one–loop calculation in themeson–baryon system, this amounts to the tree terms of order p, the loop contributionsof order p2, p3 and the counterterms of order p2 and p3. This should be kept in mindin what follows.6.2.

Non–relativistic formalismAs we saw, the fully relativistic treatment of the baryons leads to severe compli-cations in the low–energy structure of the EFT. Intuitively, it is obvious how one canrestore the one–to–one correspondence between the loop and the small momentum ex-pansion.

If one considers the baryons as extremely heavy, only the relative pion momentawill count and these can be small. The emerging picture is that of a very heavy sourcesurrounded by a cloud of light (almost massless) particles.

This is exactly the same ideawhich is used in the so–called heavy quark effective field theory methods used in heavyquark physics. To be precise, consider a meson of the type Q¯q (or ¯Qq), with Q = b, tthe heavy and q = u, d the light quark.

The light quarks and the gluons are a cloudaround the heavy source Q, they form the so–called “brown mug”. This is the pictureunderlying the so–called heavy quark symmetries in QCD [146] which allows e.g.

torelate a set of form factors to one invariant function (sometimes called the “Isgur–Wisefunction”). Therefore, it appears natural to apply the insight gained from heavy quarkEFT’s to the pion–nucleon sector.

Jenkins and Manohar [147,148,149] have given a newformulation of baryon CHPT based on these ideas. It amounts to taking the extremenon–relativistic limit of the fully relativistic theory and an expansion in powers of theinverse baryon mass.

Obviously, the relativistic theory and its extreme non–relativisticlimit are connected by a series of Foldy–Wouthuysen transformations [150].Let me first spell out the underlying ideas before I come back to the πN system.Our starting point is a free Dirac field with mass mL = ¯Ψ(i̸∂−m)Ψ(6.20)Consider the spin–1/2 particle very heavy. This allows to write its four–momentum aspµ = mvµ + lµ(6.21)with vµ the four–velocity satisfying v2 = 1 and lµ a small off–shell momentum, v ·l ≪m(for the present discussion we can set lµ = 0).

One can now construct eigenstates of thevelocity projection operator Pv = (1 + ̸v)/2 viaΨ = e−imv·x (H + h)̸vH = H ,̸vh = −h(6.22)63

which in the nucleon rest–frame vµ = (1, 0, 0, 0) leads to the standard non–relativisticreduction of a spinor into upper and lower components. Substituting (6.22) into (6.20)one findsL = ¯H(iv · ∂)H −¯h(iv · ∂+ 2m)h + ¯Hi̸∂⊥h + ¯hi̸∂⊥H(6.23)with ̸∂⊥the transverse part of the Dirac operator, ̸∂= ̸v(v · ∂) + ̸∂⊥.

Form eq. (6.23) itfollows immediately that the large component field H obeys a free Dirac equationv · ∂H = 0(6.24)modulo corrections which are suppressed by powers of 1/m.

A more elegant path integralformulation is given by Mannel et al. [151].

There is one more point worth noticing. Inprinciple, the field H should carry a label ’v’ since it has a definite velocity.

The Lorentzinvariant Lagrangian L follows from Lv = L[ ¯Hv, Hv] via a suitable integration,L =Z d3v2v0Lv(6.25)This is of importance when the incoming and outgoing baryons do not have the samevelocity. For most of the purposes to be discussed we do not need to worry about thelabel ’v’ and will therefore drop it.Let me now return to the πN system.

The reasoning is completely analogous tothe one just discussed. I follow here the systematic analysis of quark currents in flavorSU(2) of Bernard et al.

[152]. Starting from eq.

(6.11) we write Ψ in the form of eq. (6.22)and eliminate the small component field h via its equation of motion,h = 12(1 −̸v) 12 ◦mi̸D +◦gA2 ̸uγ5H + O(1/ ◦m2)(6.26)This leads toL(1)πN = ¯Hiv · D + ◦gAS · uH+ 12 ◦m¯Hi̸D +◦gA2 ̸uγ51 −̸v2i̸D +◦gA2 ̸uγ5H + O(1/ ◦m2)(6.27)As advertised, the nucleon mass term has disappeared.

Furthermore, any Dirac bilinear¯ΨΓµΨ (Γµ = 1, γµ, γ5, . .

.) can be expressed in terms of the velocity vµ and the spin–operator 2Sµ = iγ5σµνvν.

It obeys the relationsS · v = 0, S2 = −34,Sµ, Sν= 12vµvν −gµν, [Sµ, Sν] = iǫµναβvαSβ(6.28)Using the convention ǫ0123 = −1, we can rewrite the standard Dirac bilinears as [147,152]:¯HγµH = vµ ¯HH, ¯Hγ5H = 0, ¯Hγµγ5H = 2 ¯HSµH¯HσµνH = 2ǫµναβvα ¯HSβH, ¯Hγ5σµνH = 2i(vµ ¯HSνH −vν ¯HSµH)(6.29)64

Therefore, the Dirac algebra is extremely simple in the extreme non–relativistic limit.The disappearance of the nucleon mass term to leading order in 1/ ◦m now allows for aconsistent chiral power counting. The tree level Lagrangian is of order O(q) and one–loop diagrams contribute to order O(q3), with q now a genuine small momentum (forthe baryons, only the small lµ counts).

The power counting scheme is discussed in detailby Jenkins and Manohar [149] based on the chiral quark model approach of Manoharand Georgi [20]. In essence, since all momenta can be considered small compared to thechiral scale Λ ≃Mρ one has a scheme as in the meson sector, with the main differencethat odd powers in q are allowed here.

An important point is that all vertices now consistof a string of operators with increasing powers in 1/m. For the calculation of one–loopdiagrams, one needs, however, only the mass independent propagator and vertices, wehave for exampleNucleon propagator :S(ω) =iv · l + iη , η > 0Photon −nucleon vertex :ie1 + τ32ǫ · v + O(1/ ◦m)Pion −nucleon vertex :(◦gA/F)τ aS · q + O(1/ ◦m)(6.30)with ω = v·l.

For the contact terms, one therefore has to go further in the 1/m expansionto achieve the same accuracy. This means that in case of a one–loop calculation one hasto take into account all local contact terms which contribute to the order the loops do.A consequence is that the effective Lagrangian written down by Jenkins and Manohar[147] is not complete since it only contains terms with one derivative or one quarkmass insertion, i.e.

L(1)πN and parts of L(2)πN. However one should also account for L(3)πN.To make this statement more transparent, consider the isovector anomalous magneticmoment of the nucleon.

The isovector–vector quark current can be expressed in termsof two form factors. In the heavy mass limit, this is most conveniently done in the Breitframe where the photon transfers no energy since it allows for a unique translation ofLorentz–invariant matrix elements into non–relativistic ones.

For the case at hand, thisgives< N(p′)|¯qγµτ3q|N(p) >=F V1 (t) +t4m2NF V2 (t)vµ ¯Hτ3H+1mNF V2 (t) + F V1 (t) ¯H[Sµ, S · (p′ −p)]τ3H . (6.31)Consider now the Pauli form factor F V2 (t).

At zero momentum transfer, it gives theisovector anomalous magnetic moment, F V2 (0) = κV = κp −κn = 3.71. The calculationof F V2 (t) is straightforward [152], at t = 0 one findsκV = c′6 −◦g2AM ◦m4πF 2(6.32)65

where the second term on the r.h.s. of (6.32) is the one–loop contribution (which is non–analytic in the quark mass since M ∼√ˆm [155]) and the constant c′6 has to be identifiedwith the anomalous isovector magnetic moment in the chiral limit.

It originates from acounterterm of L(2)πN,L(2)πN = c′6 + 14 ◦m ǫµναβvµ ¯HSνf +αβH(6.33)where the field strength tensor f +αβ is defined via f +αβ = u†Fαβu† + uFαβu with Fαβthe conventional photon fields strength tensor. Choosing c′6 = 5.62, one reproduce theempirical value of κV .

This value of ◦κV = c′6 is close to what one would estimate if onegenerates the contact term (6.33) from ρ–exchange since the tensor coupling of the ρ isabout κρ ≃6. Other observables and the interplay between the loop and countertermcontributions are discussed in ref.

[152].Another topic is the matching to the relativistic formalism. This has been addressedin ref.[152].

The underlying assumption is that the extreme non–relativistic formulationshould match to the relativistic one (this point of view is not shared by everybody). Inthat case, one can derive matching conditions between the various low–energy constants.This allows one to use previously determined values of these coefficients also in thenon–relativistic approach.

In fact, these low–energy constants depend on the scale ofdimensional regularization, called λ here. To be specific, consider the wave–functionrenormalization (Z–factor) of the nucleon.

It is defined by the residue of the propagatorat the physical mass pole,ip · v −◦m −Σ(ω) →iZNp · v −mN(6.34)with Σ(ω) the nucleon self–energy in the one–loop approximation. One findsZN = 1 + Σ′(0) = 1 −3◦g2AM 232π2F 23 ln Mλ + 1−4M 2F 2 b′r12(λ)(6.35)where the coupling constant b′12 has eaten up the infinity from the loop [142,152].

If onecompares (6.35) to the expression of GSS [142] expanded in powers of µ = M/ ◦m, onefindsZrelN = 1 −3◦g2AM 232π2F 2ln Mλ + 2 ln µ + 2−4M 2F 2 br12(λ) + O(µ3)(6.36)Therefore, the heavy mass and the relativistic theory only match if one setsλ = ◦mb′12 = b′r12(λ) + γ′b12L,γ′b12 = −98◦g2A(6.37)66

The result λ = ◦m is not quite unexpected since one has integrated out the field h of mass2 ◦m. Therefore, if one chooses to retain only the leading non–analytic contributions fromthe loops (as it is often done), one has to set λ = ◦m.

However, as stressed many timesbefore, an accurate CHPT calculation at a given order has to account for all loop andcounterterm contributions to the order one is working. A more detailed discussion of theinterplay between the relativistic and the heavy mass formulation is given in ref.

[152].One particular advantage of the heavy mass formulation is the fact that it is veryeasy to include the baryon decuplet, i.e. the spin–3/2 states.

This has been done infull detail by Jenkins and Manohar [148]. The inclusion of the ∆(1232) is motivatedmostly by the following argument: The N∆mass–splitting m∆−mN is only abouttwice as much as the pion mass, so that one expects significant contributions from thisclose–by resonance (the same holds true for the full decuplet in relation to the octet).This expectation is borne out in many phenomenological models.

However, it shouldbe stressed that if one chooses to include this baryon resonance (or the full decuplet),one again has to account for all terms of the given accuracy one aims at, say O(q3) in aone–loop calculation. This tends to be overlooked in the presently available literature.Furthermore, the mass difference m∆−mN does not vanish in the chiral limit thusdestroying the consistent power counting (as it is the case with the baryon mass inthe relativistic formalism discussed in section 6.1).

I will not consider the full baryonoctet coupled to the decuplet but will only focus on the physics of the πN∆system. Thedecuplet fields are described by Rarita–Schwinger spinor ( ˜T µ)abc with a, b, c ∈{1, 2, 3}.This spinor contains both spin–1/2 and spin–3/2 components.

The spin–1/2 pieces areprojected out by use of the constraint γµ ˜T µ = 0. Under SU(3) × SU(3) ˜T µ transformsas ( ˜T µ)abc →(K)da(K)eb(K)fc( ˜T µ)def and one defines a velocity dependent field via˜T µ = e−im∆v·x (T + t)µ(6.38)with m∆the ∆(1232)mass (in general one should use the average decuplet mass).

Interms of the physical states we have T111 = ∆++, T112 = ∆+/√3, T122 = ∆0/√3,T222 = ∆−. The effective ∆Nπ Lagrangian to leading order readsL(1)∆Nπ = −i ¯T µv · DTµ + ∆¯T µTµ + 3◦gA2√2( ¯T µuµH + ¯HuµT µ)(6.39)with ∆= m∆−mN.

Clearly one is left with some residual mass dependence fromthe N∆–splitting (notice that the average octet–decuplet splitting is smaller than ∆).In the language of ref. [148] we have set C = 3◦gA/2 which is nothing but the couplingconstant relation g∆Nπ = 3gπN/√2 = 1.89.

Empirically this relation is fulfilled withina few per cent. From the width of the decay ∆→Nπ one has C = 1.8 [149], consistentwith the value given before (if one uses the full decuplet the value of C reduces to 1.5).The propagator of the spin–3/2 fields readsS∆(ω) = ivµvν −gµν −43SµSνω −∆(6.40)67

For all practical purposes, it is most convenient to work in the rest–frame vµ = (1, 0, 0, 0).In that case, one deals with the well–known non–relativistic isobar model which is dis-cussed in detail in the monograph by Ericson and Weise [154]. We will come back tosome calculations involving the ∆(1232) (or the full decuplet) in the following section.A last comment on the heavy fermion EFT is necessary.

While it is an appealingframework, one should not forget that the nucleon (baryon) mass is not extremely large.Therefore, one expects significant corrections from 1/m suppressed contributions tomany observables.This will become more clear in the discussion of threshold pionphoto– and electroproduction.It is conceivable that going to one–loop order O(q3)is not sufficient to achieve a very accurate calculation.This means that even morethan in the meson sector higher–loop calculations should be performed to learn aboutthe convergence of the chiral expansion. Ultimately, one might want to include moreinformation in the unperturbed Hamiltonian.

At present, it is not known how to dothat but it should be kept in mind.6.3. Photo–nucleon processesIn this section, I will be concerned with reactions involving photons, nucleons andpions.

There has been much impetus from the experimental side in the recent yearsthrough precise measurements of the nucleons electromagnetic polarizabilities or therecent data on threshold pion photo– and electroproduction.Let us first consider nucleon Compton scattering at low energies. The energy ex-pansion of the spin–averaged Compton amplitude in the rest frame takes the formT(γN →γN) = −e2Z24πm + ¯α ω′ω⃗ǫ′ ·⃗ǫ + ¯β (⃗ǫ′ ×⃗k′) · (⃗ǫ ×⃗k) + O(ω4)(6.41)with (ω,⃗k,⃗ǫ) and (ω′,⃗k′,⃗ǫ′) the frequencies, momenta and polarization vectors of the in-coming and outgoing photon, respectively.

The first term, which is energy–independent,is nothing but the Thomson scattering amplitude as mandated by gauge invariance.It constitutes a low–energy theorem as the photon energy tends to zero. At next–to–leading order in the energy expansion, the photon probes the non–trivial structure ofthe spin–1/2 particle (here, the nucleon) it scatters off.

This information is encodedin two structure constants, the so–called electric (¯α) and magnetic (¯β) polarizabilities.These have been determined rather accurately over the last years [155]¯αp = 10.8 ± 1.0 ± 1.0 , ¯βp = 3.4 ∓1.0 ∓1.0¯αn = 12.3 ± 1.5 ± 2.0 , ¯βn = 3.1 ∓1.6 ∓2.0(6.42)all in units of 10−4 fm−3 which I will use throughout. The errors in (6.42) are anticor-related since one imposes the well–established dispersion sum rule results (¯α + ¯β)p =14.2 ± 0.3 and (¯α + ¯β)n = 15.8 ± 0.5[156].

So the proton and the neutron behave essen-tially as electric dipoles and their respective sum of the polarizabilities is approximately68

equal.For calculating the em polarizabilities, it is most convenient to consider thespin–averaged Compton tensor in forward direction [157,158]θµν = 14 Tr {(̸p + m)Zd4x eik·x < N(p)|T Jemµ (x)Jemν(0)|N(p) > }= e2 {gµν A(s) + kµkν B(s) + (pµkν + pνkµ) C(s) + pµpν D(s)}(6.43)The pertinent Mandelstam variables are s = (p + k)2 and u = 2m2 −s since t = 0 inforward direction. Gauge invariance (kµΘµν = 0) and u ↔s crossing symmetry reducethe number of independent scalar functions in (6.43) to two.

It is straightforward toread offthe polarizabilities¯α + ¯β = −e2m2πA′′(m2) , ¯β = −e24πm B(m2)(6.44)where the prime denotes differentiation with respect to s. The non–renormalizationtheorem of the electric charge leads to the additional constraint A(m2) = Z2.To one–loop in CHPT, the em polarizabilities are pure loop effects. This means thatthere are no contributions from the contact terms L(2,3)πN .

Therefore, no undeterminedlow–energy constants enter the final results and one tests the loop structure of chiralQCD. In the relativistic formalism, the calculation of ¯αp,n and ¯βp,n has been performedby Bernard et al.in refs.

[157,158].Although one has to evaluate 52/22 Feynmandiagrams for the proton/neutron, the final results are simple one dimensional integralsdue to the particular kinematics. In chiral SU(2), the expansion of the polarizabilitiesin powers of µ = Mπ/m (ratio of the pion and the nucleon mass) takes the form¯αp = C5π2µ + 18 lnµ + 332 + O(µ)¯αn = C5π2µ + 6 lnµ −32 + O(µ)¯βp = C π4µ + 18 lnµ + 632 + O(µ)¯βn = C π4µ + 6 lnµ + 52 + O(µ)(6.45)with C = e2g2πN/192π3m3 ≃0.26 · 10−4 fm3.To leading order, the polarizabilitiesdiverge as 1/Mπ as Mπ tends to zero.

This is an expected result since the pion cloudbecomes long–ranged in the chiral limit, i.e. the Yukawa suppression factors turn intoa simple power law fall–off.

In fact, in the heavy mass formalism, it is only this mostsingular term which results from the one–loop calculation [152]. Working in the Coulombgauge ǫ · v = 0, it is easy to filter out the diagrams which lead to this singularity.69

These diagrams were also found in the relativistic formalism, however, after tediouscalculations. The most singular term reproduces the data fairly well¯αp = ¯αn = 5e2g2πN384π2m21Mπ= 13.6¯βp = ¯βn = ¯αp/10 = 1.4(6.46)i.e.

(¯α + ¯β)p = (¯α + ¯β)n = 14.9 close to the empirical values and, furthermore, ¯αp,n ≫¯βp,n. Some of the 1/m corrections modifying these leading order results are resummedin the relativistic calculation, they tend to decrease the polarizabilities and lead to toosmall values for the sums [157,158].However, there is also the contribution from the∆(1232) which is important for ¯βp,n in many phenomenological models.

Clearly, the1/m suppressed terms lead to isospin breaking (¯αp ̸= ¯αn, ¯βp ̸= ¯βn). In a recent paper[159] the ln µ–term in (6.45) was criticised as incorrect since the nucleon mass appears inthe logarithm.

This is, however, a false statement – from the discussion of the matchingcondition λ = m it should be clear that such terms are indeed possible. Clearly, theyare not the whole story at order q4.

When one extends the calculation to flavor SU(3),one has to consider pion and kaon loops. This was for the first time done in ref.

[160],where also predictions for the electromagnetic polarizabilities of the hyperons are given.For typical values of the SU(3) F and D couplings (see appendix B), the kaon loopstend to increase ¯αp over ¯αn by about 15 per cent. A similar observation was also madein ref.[159].

In that paper, in addition the contribution from the decuplet (∆, Σ, Σ⋆) inthe intermediate states were considered (the ∆–loop contribution had previously beengiven by Kaiser [161]). The authors of ref.

[159] finds ¯αp = 14.1, ¯αn = 13.4, ¯βp = 0.2 and¯βn = 0.1 not too different from the leading order results (6.46) (clearly, the magneticpolarizabilities are most affected by the decuplet contribution). However, unrealisticallysmall values for the F and D coupling constants are used in this calculation.

Whathas to be done here is to perform a full–fledged calculation to include all terms up–to–and–including O(q4). This is possible in the heavy mass formalism and is the only wayto get a clean handle at the terms modifying the leading order (q3) result (6.46).

Inref. [152] a prediction was also made for the spin–dependent Compton amplitude f2(ω)at low ω and in ref.

[162], the small momentum expansion of the Drell–Hearn–Gerasimovsum rule was investigated. These predictions become relevant when the scattering ofpolarized photons on polarized nucleons will be performed.For further details, thereader is referred to refs.

[152,162].Next, let me consider threshold pion photo– and electroproduction. In the thresholdregion, i.e.

when the real photon has just enough energy to produce the pion at rest, thedifferential cross section for the process γ(k) + p(p1) →π0(q) + p(p2) can be expressedin terms of the electric dipole amplitude E0+, dσ/dΩ= (E0+)2 as ⃗q tends to zero.This multipole is of particular interest since in the early seventies a low–energy theorem(LET) for neutral pion production was derived [163]. The recent measurements at Saclayand Mainz [164] seemed to indicate a gross violation of this LET, which predicts E0+ =70

−2.3 · 10−3/Mπ+ at threshold.However, the LET was reconsidered (and rederived)and the data were reanalyzed, leading to E0+ = (−2.0 ± 0.2) · 10−3/Mπ+ in agreementwith the LET prediction. These developments have been subject of a recent reviewby Drechsel and Tiator [165].Therefore, I will focus here on the additional insightgained from CHPT calculations.

In fact, the LET for neutral pion photoproduction atthreshold is an expansion in powers of µ = Mπ/m ∼1/7 and predicts the coefficientsof the first two terms in this series, which are of order µ and µ2, respectively, in termsof measurable quantities like the pion–nucleon coupling constant gπN, the nucleon massm and the anomalous magnetic moment of the proton, κp. In ref.

[166] it was, however,shown that a certain class of loop diagrams modifies the LET at next–to–leading orderO(µ2). This analysis was extended to threshold pion electroproduction γ⋆p →π0p inref.[167].

In the electroproduction case, there is another S–wave multipole called L0+due to the longitudinal coupling of the virtual photon to the nucleon. The resultingexpressions for E0+ and L0+ to order O(µ2) and O(ν), with ν = −k2/m2 > 0 (k2 < 0in the electroproduction reaction) read (notice that µ is of order q whereas ν is of orderq2):E0+ = C−µ + µ212(3 + κp) +m24πF 2πξ2(−ν/µ2)−ν12(1 + κp) +m28πF 2π2ξ2(−ν/µ2) −ξ1(−ν/µ2)+ O(q3)L0+ = C−µ + µ232 +m28πF 2πξ1(−ν/µ2)−ν2 + +O(q3)(6.47)with C = egπN/8πm = 0.17 GeV−1.

The transcendental functions ξ1,2(α), α = ν/µ2can not be further expanded in µ and ν separately. They are given by:ξ1(α) =11 + α +α2(1 + α)3/2π2 + arcsinα2 + αξ2(α) =3α4(1 + α)2 + 4 + 4α + 3α28(1 + α)5/2π2 + arcsinα2 + α(6.48)At the photon point k2 = 0, one has ν = 0 and ξ1(0) = 1, ξ2(0) = π/4.

Therefore, toorder µ2 one has a term ∼(m/4Fπ)2 which modifies the LET of ref.[163]. To order µ2and ν, the form (6.48) is the correct one as given by QCD.

The origin of these novelcontributions proportional to ξ1,2(α) can most easily be traced back in the heavy massformalism. In ref.

[152] it was shown that working in the Coulomb gauge and the protonrest frame, only four diagrams are non–vanishing for π0 threshold production. Theseare the so–called triangle and rescattering diagrams and their crossed partners.

Thecontribution of the triangle diagram is non–analytic in the quark mass, ∆A1 ∼Mπ ∼√ˆm [166].This non–analyticity is generated by the loop integral associated to this71

Fig. 13:a) Total cross section for γp →π0p from the one–loop calculation ofRef.[169].

The tree and one-loop results are indicated by the dashed and solidline, respectively.The data are from the Saclay (closed circles) and Mainz(open circles) groups [164].b) Same with the inclusion of isospin–breakingin comparison with the Mainz data. Notice the weak cusp at π+n threshold(Eγ = 151.4 MeV).

After the π+n threshold, the predicted total cross sectionagrees with the one given in Fig.13a.diagram, it develops a logarithmic singularity in the chiral limit. Therefore, the naivearguments based on s ↔u crossing which lead to ∆A1 ∼M 2π are invalid.

To see thiseffect, one has to calculate loop diagrams (this was known to Vainsthein and Zakharov[163], but they unfortunately calculated the harmless rescattering diagram). Notice alsothat the anomalous magnetic moment κp is generated by loops and counterterms inCHPT.

It would therefore be surprising if that would be the only such contribution –and it is not. As a final comment on eq.

(6.47) let me point out that the LETs containno photon–nucleon form factors [168]. These are build up order by order in the chiralexpansion and to order O(q2), one only sees their leading terms.

In ref. [169], a completeanalysis of threshold pion photoproduction was given based on the relativistic formalism.72

This is necessary since in the static limit one only gets the terms including order µ2 inE0+. A quick look at (6.47) reveals that the coefficient of the µ2 term is large and sothe leading order term ∼µ is cancelled.

This means that the expansion in µ is slowlyconverging and one has to go beyond the extreme non–relativistic limit. The total crosssection for γp →π0p in the threshold region is shown in fig.13a.

It agrees reasonablywell with the data. In that paper, one can also find a prediction for the cross sectionof γn →π0n.This process is even more sensitive to the chiral symmetry breakingsince for the electric dipole amplitude at threshold, E0+ ∼O(µ2).

A measurement ofthis reaction is intended using a deuteron target [170]. In the one–loop approximationused in these calculations, one has no isospin–breaking and therefore Mπ0 = Mπ+ andmp = mn.

To study some of the effects of isospin–breaking, in ref. [171] a calculationwas presented in which the physical masses for the charged and neutral pions as wellas for the proton and neutron were used (this is not a complete calculation since atthis order there are many other effects).In that case, one has two thresholds, π0pand π+n, separated by ≃5 MeV.

The cross section in this energy regime is shown infig.13b. It is clearly improved compared to the one–loop prediction.

This indicates thata calculation beyond the isosymmetric one–loop approximation should be performed.Very recently, new accurate data very close to threshold have become available for theelectroproduction process γ⋆p →π0p [172]. In particular, the S–wave cross section a0,which is a particular combination of E0+ and L0+, was analyzed (for the first time inthis kinematical regime).

A one loop CHPT calculation was performed by Bernard etal. [173] and it was shown that loop effects are indeed necessary to explain the data asshown in fig.14.Fig.

14:The S–wave component of the neutral pion electroproduction crosssection, calculated from CHPT (solid line) and PV (dotted line) are compared.The kinematics is W = 1074 MeV and ǫ = 0.58 [173]. The data extracted inref.

[172] are also shown.73

Consider now the photoproduction of charged pions. In that case, there exists theLET due to Kroll and Ruderman [174] for the electric dipole amplitude (for γp →π+nand γn →π−p).

The predictions are in agreement with the data. The loop correctionsare much more tame than in the case of neutral pion production, they start at orderO(µ2, µ2 ln µ2) whereas the leading term is of order one.

In the case of charged pionelectroproduction, an interesting observation was made in ref.[175]. The starting point isthe venerable LET due to Nambu, Luri´e and Shrauner [176] for the isospin–odd electricdipole amplitude E(−)0+ in the chiral limit,E(−)0+ (Mπ = 0, k2) = egA8πFπ1 + k26 r2A + k24m2 (κV + 12) + O(k3)(6.49)Therefore, measuring the reactions γ⋆p →π+n and γ⋆n →π−p allows to extract E(−)0+and one can determine the axial radius of the nucleon, rA.

This quantity measures thedistribution of spin and isospin in the nucleon, i.e. probes the Gamov–Teller operator⃗σ·⃗τ.

A priori, the axial radius is expected to be different from the typical electromagneticsize, rem ≃0.8 fm. It is customary to parametrize the axial form factor GA(k2) by adipole form, GA(k2) = (1 −k2/M 2A)−2 which leads to the relation rA =√12/MA.

Theaxial radius determined from electroproduction data is typically rA = 0.59 ± 0.04 fm[177] whereas (anti)neutrino-nucleon reactions lead to somewhat larger values, rA =0.65 ± 0.03 fm. This discrepancy is usually not taken seriously since the values overlapwithin the error bars.

However, it was shown in ref. [175] that pion loops modify theLET (6.49) at order k2 for finite pion mass.

In the heavy mass formalism, the coefficientof the k2 term reads16r2A +14m2 (κV + 12) +1128F 2π(1 −12π2 )(6.50)where the last term in (6.50) is the new one.This means that previously one hadextracted a modified radius, the correction being 3(1 −12/π2)/64F 2π ≃−0.046 fm2.This closes the gap between the values of rA extracted from electroproduction andneutrino data. It remains to be seen how the 1/m suppressed terms will modify theresult (6.50).

Such investigations are underway.6.4. Baryon masses and the σ–termIn this section, I will mostly be concerned with some recent work on the so–calledpion–nucleon σ–term related to low–energy πN scattering.

As an entr´ee, however, itis mandatory to briefly discuss the chiral expansion of the baryon masses. As we willsee later, the σ–term is nothing but a nucleon mass shift related to the finiteness of thequark masses.74

Gasser [55] and Gasser and Leutwyler [3] were the first to systematically investigatethe baryon masses at next–to–leading order. The quark mass expansion of the baryonmasses takes the formmB = m0 + αM + βM3/2 + γM2 + .

. .

(6.51)The constant m0 reminds us that the baryon masses do not vanish in the chiral limit.The non–analytic piece proportional to M3/2 was first observed by Langacker and Pagels[141]. If one retains only the terms linear in the quark masses, one obtains the Gell-Mann–Okubo relation mΣ + 3mΛ = 2(mN + mΞ) (which is fulfilled within 0.6 percent in nature) for the octet and the equal spacing rule for the decuplet, mΩ−mΞ∗=mΞ∗−mΣ∗= mΣ∗−m∆(experimentally, one has 142:145:153 MeV).

However, to extractquark mass ratios from the expansion (6.51), one has to work harder. This was done inrefs.[3,55].

The non–analytic terms were modelled by considering the baryons as staticsources surrounded by a cloud of mesons and photons – truely the first calculation inthe spirit of the heavy mass formalism. The most important result of this analysis wasalready mentioned in section 3.2, namely that the ration R = ( ˆm −ms)/(mu −md)comes out consistent with the value obtained from the meson spectrum.

Jenkins [179]has recently repeated this calculation using the heavy fermion EFT of refs. [147,148,149],including also the spin–3/2 decuplet fields in the EFT.

She concludes that the successof the octet and decuplet mass relations is consistent with baryon CHPT as long as oneincludes the decuplet. Its contributions tend to cancel the large corrections from thekaon loops like M 2K ln M 2K.

The calculation was done in the isospin limit mu = md = 0so that nothing could be said about the quark mass ratio R. One would like to see sucha refined analysis.Let me now turn to the πN σ–term. It is defined viaσ =ˆm2m < p|¯uu + ¯dd|p >= ˆm∂m∂ˆm(6.52)making use of the Feynman–Hellmann theorem and the proton state |p > is normalizedvia < p′|p >= 2(2π)3p0δ(⃗p −⃗p′).

The quantity σ can be calculated from the baryonspectrum. To leading order in the quark masses, one findsσ =ˆmms −ˆmmΞ + mΣ −2mN1 −y+ O(M3/2)y =2 < p|¯ss|p >< p|¯uu + ¯dd|p >(6.53)where y is a measure of the strange quark content of the proton.

Setting y = 0 assuggested by the OZI rule, one finds σ = 26 MeV. However, from the baryon mass75

analysis it is obvious that one has to include the O(M3/2) contributions and estimatethe O(M2) ones. This was done by Gasser [55] leading toσ = 35 ± 5 MeV1 −y=σ01 −y(6.54)However, in πN scattering one does not measure σ, but a quantity called Σ defined viaΣ = F 2π ¯D+(ν = 0, t = 2M 2π)(6.55)with the bar on D denoting that the pseudovector Born terms have been subtracted,¯D = D −Dpv.

The amplitudes D± are related to the more conventional πN scatteringamplitudes A± and B± via D± = A± + νB±, with ν = (s −u)/4m. The superscript’±’ denotes the isospin (even or odd).

D is useful since it is related to the total crosssection via the optical theorem. The kinematical choice ν = 0, t = 2M 2π (which lies inthe unphysical region) is called the Cheng–Dashen point [180].

The relation between Σand the πN scattering data at low energies is rather complex, see e.g. H¨ohler [181] fora discussion or Gasser [182] for an instructive pictorial (given also in ref.[144]).

Basedon dispersion theory, Koch [183] found Σ = 64 ± 8 MeV (notice that the error onlyreflects the uncertainty of the method, not the one of the underlying data). Gasser etal.

[184] have recently repeated this analysis and found Σ = 60 MeV (for a discussionof the errors, see that paper). There is still some debate about this value, but in whatfollows I will use the central result of ref.[184].

Finally, we have to relate σ and Σ. Therelation of these two quantities is based on the LET of Brown et al. [185] and takes thefollowing form at the Cheng–Dashen point:Σ = σ + ∆σ + ∆R∆σ = σ(2M 2π) −σ(0)(6.56)∆σ is the shift due to the scalar form factor of the nucleon, < p′| ˆm(¯uu + ¯dd)|p >=¯u(p′)u(p)σ(t), with t = (p′ −p)2 and ∆R is related to a remainder not fixed by chiralsymmetry.

The latter was found to be very small by GSS [142], ∆R = 0.4 MeV. Theone–loop calculation of ref.

[142] for ∆σ gave 4.6 MeV, and in the heavy mass formulationone finds 7.4 MeV [152]. Adding up the pieces, this would amount to y ≈0.3 .

. .0.4, i.e.a large strange quark content in the nucleon.

However, since one is dealing with strongS–wave ππ and πN interactions, the suspicion arises that the one–loop approximationis not sufficient, as already discussed in section 4.4. Therefore, Gasser et al.

[184,186]have performed a dispersion–theoretical analysis tied together with CHPT constraintsfor the scalar form factor σ(t). The resulting numerical value is∆σ = (15 ± 0.5) MeV(6.57)which is a stunningly large correction to the one–loop result.

If one parametrizes thescalar form factor as σ(t) = 1 + r2S t + O(t2), this leads to r2S = 1.6 fm2, substantially76

larger than the typical electromagnetic size. This means that the scalar operator ˆm(¯uu+¯dd) sees a more extended pion cloud.

Notice that for the pion, a similar enhancement ofthe scalar radius was already observed, (r2S/r2em)π ≃1.4 [13]. Putting pieces together,one ends up with σ = 45 ± 8 MeV [184] to be compared with σ0/(1 −y) = (35 ±5) MeV/(1 −y).

This means that the strange quarks contribute approximately 10 MeVto the σ–term and thus the mass shift induced by the strange quarks is ms < p|¯ss|p >≃(ms/2 ˆm) · 10 MeV ≃130 MeV. This is consistent with the estimate made in ref.

[187]based on the heavy mass formalism including the decuplet fields.The effect of thestrange quarks is not dramatic. All speculations starting from the first order formula(6.53) should thus be laid at rest.

The lesson to be learned here is that many smalleffects can add up constructively to explain a seemingly large discrepancy like Σ −σ0 ≈σ0. What is clearly needed are more accurate and reliable low–energy pion–nucleonscattering data to further pin down the uncertainties.

For an update on these issues,see Sainio [188].6.5. Non-leptonic hyperon decaysSince the early days of soft–pion techniques, the non–leptonic hyperon decays havebeen studied using EFT methods (see e.g.the monograph [189]).There are sevensuch decays, Λ →π0n , Λ →π−p , Σ+ →π+n , Σ+ →π0p , Σ−→π−n , Ξ0 →π0Λand Ξ−→π−Λ.

These are compactly written as Hba, where H denotes the decayinghyperon, a = −, 0, + gives the charge of the outgoing pion and b = −, 0, + denotes thecharge of the decaying hyperon. For example, Λ00 stands for Λ →π0n.

These decaysare produced by the strangeness–changing ∆S = 1 weak Hamiltonian proportional to¯uγµ(1+γ5)s ¯dγµ (1+γ5)u (modulo QCD corrections). Under SU(3)×SU(3) this operatortransforms as (8L, 1R) ⊕(27L, 1R).

Experimentally, the octet piece dominates the decayamplitudes. Therefore, we will neglect the 27–plet contributions in what follows.

Theinvariant matrix elements take the formM(Hba) = GF M 2π¯uB′A(S)(Hba) + γ5A(P )(Hba)uB(6.58)where uB denotes the spinor of baryon B and ¯uB = u+Bγ0.The S–wave decay isparity–violating whereas the P–wave decays conserve parity. Isospin symmetry of thestrong interactions implies three relations among the amplitudes, separately for theS– and the P–waves.

These are M(Λ0−) = −√2M(Λ00) , M(Ξ−−) = −√2M(Ξ00) and√2M(Σ+0 ) + M(Σ−−) = M(Σ++). Thus, only four amplitudes are independent.To leading order in the effective theory, ∆S = 1 non–leptonic Hamiltonian reads[20,21]Heff(∆S = 1) = a Tr ¯B{u+hu, B}+ b Tr ¯B[u+hu, B]+ h.c.(6.59)where the explicit expression for the baryon field B is given in appendix B and the traceruns over the flavor indices.

The SU(3) matrix h is a spurion field to project out theoctet component, hji = δj2δ3i . From (6.59) it follows for the S–waves that A(S)(Σ++) = 0,77

A(S)(Σ−−) = (a −b)/F, A(S)(Λ0−) = −(a + 3b)/√6F and A(S)(Ξ−−) = (3b −a)/√6F.These can be combined to give the Lee–Sugawara relation [190],A(S)(Λ0−) + 2A(S)(Ξ−−) +r32A(S)(Σ−−) = 0(6.60)which is experimentally quite well fulfilled (see below). No such simple relation can bederived for the P–wave amplitudes since these always involve pole diagrams in whichthe baryon changes strangeness before or after pion emission.

It is worth noting thatto leading order, both the S– and the P–wave decay amplitudes are independent of thestrange quark mass (setting mu = md = 0). Furthermore, it is not possible to achieveeven a descent fit for the P–wave amplitudes once the parameters a and b in (6.59) arefixed to reproduce the S–waves.

This together with a possible solution involving higherdimensional operators is discussed in some detail in refs. [20,21].Naturally, the question arises what the chiral corrections to these decays are.

Bi-jnens et al. [191] were the first to address this issue.

Apart from the standard lowestorder chiral meson-baryon Lagrangian (see appendix B) they used for the symmetrybreaking the following term:L(2) = a1 Tr [M(U+U †)]+b1 Tr [ ¯B(u†Mu†+uMu)B]+b2 Tr [ ¯BB(u†Mu†+uMu)] (6.61)and set M = diag(0, 0, ms), i.e. only kaon loops contribute.

The parameters a1 , b1and b2 were determined from the mass spectrum. They worked in the one–loop approx-imation keeping only the non–analytic terms of the type ms ln ms ∼M 2K ln M 2K.

Nolocal contact terms were considered. This, of course, introduces a scale dependence,the subtraction point was chosen to be µ = 1 GeV.

Tuning the parameters a and b(6.59) to get a best fit to the S–wave amplitudes, the following results emerged. Theone–loop correction was zero for A(S)(Σ++), of the order of 30 per cent for A(S)(Σ−−) andA(S)(Λ0−) and 75 per cent for A(S)(Ξ−−).

The resulting S–wave predictions are no longersatisfactorily agreeing with the data and the Lee–Sugawara relation breaks down. In theP–waves, the corrections are even larger, in two cases much bigger than the tree result.None of the A(P )(Hba) can be reproduced, the discrepancies to the empirical numbers arelarge.

This calculation sheds some doubt on the validity of the chiral expansion in thethree flavor sector. Jenkins [192] has recently reinvestigated this topic.

Her calculationdiffers in various ways from the one of ref.[191]. First, she uses the heavy mass formalismand also includes the spin–3/2 decuplet in the effective theory.

Second, she accounts forwave–function renormalization and third, includes another higher derivative term. Thisis nothing but the first term in (4.26) with λ6 substituted by the spurion h. The reasonto include this term is the strong ∆I = 1/2 enhancement.

The other assumptions arethe same (no counter terms and only the leading non–analytic pieces from the loops areretained. Also, mu = md = 0 is assumed).

The main result of this calculation is thelarge cancellation between the octet and decuplet pieces in the loops (already observed78

in the baryon mass calculation [179]), therefore the total loop contribution is consider-ably reduced. In units of GF M 2π, the results of ref.

[192] read: A(S)(Σ++) = −0.09 (0.06±0.01) , A(S)(Σ+0 ) = −1.41 (−1.43 ± 0.05) , A(S)(Σ−−) = 1.90 (1.88 ± 0.01) , A(S)(Λ0−) =1.44 (1.42 ± 0.01) , A(S)(Λ00) = −1.02 (−1.04 ± 0.01) , A(S)(Ξ−−) = −2.04 (−1.98 ± 0.01)and A(S)(Ξ00) = 1.44 (1.52 ± 0.02). Good agreement between theory and experiment isfound.

Furthermore, the correction to the Lee–Sugawara relation is small. In case ofthe P–wave amplitudes, the central parameters do not lead to satisfactory descriptionof the data.

Indeed, the A(P ) are very sensitive to some of the input parameters. Whatis more important is the observation that in the case of the P–waves, the tree level pre-diction consists of two terms which tend to cancel to a large extent.

This suppressionof the SU(3) symmetric amplitudes effectively enhances the loop corrections, the termsof the type ms ln ms are as important as the tree contributions (also they are nominallysuppressed). This means that in the P–wave amplitudes one has large SU(3) viola-tion, but not a breakdown of the chiral expansion.

To get the same accuracy on theseSU(3)–violating pieces as on the SU(3)–symmetric ones which dominate the S–waveamplitudes, one would have to work much harder. Clearly, the last word is not spokenhere since none of the calculations performed so far accounts for all terms at one looporder, the leading non–analytic corrections are just one part of the whole story.Neufeld [193] has recently considered the strangeness–changing radiative decaysΣ+ →pγ, Ξ−→Σ−γ, Σ0 →nγ, Λ →nγ, Ξ0 →Λγ and Ξ0 →Σ0γ in the relativisticformalism.

The general form of the amplitude Bb →Ba + γ (a, b = 1, . .

., 8) takes theformMab = ¯uaiσµνǫµ(pa −pb)ν (Aab + Babγ5)ub(6.62)The parity–conserving amplitude Aab can be expressed in terms of the magnetic mo-ments and therefore the corresponding counterterms can be fixed.The situation isdifferent for the counterterms related to the parity–violating amplitude Bab. They arenot known but restricted by CPS–symmetry [103].

The general structure of these coun-terterms is discussed in ref.[193]. It is argued that the one–loop contributions togetherwith the lowest order contact terms (the pieces which are left for vanishing momenta) arenot sufficient to explain the existing data.

However, the imaginary parts are uniquelydetermined by the loop graphs. In particular, for the decays Λ →nγ and Ξ−→Σ−γ onecan give a prediction for the asymmetry parameter αab = 2Re(AabB∗ab)/(|A2ab| + |B2ab|)based on the recent measurements for the pertinent branching ratios.

These constraintslead to−0.8 ≤αnΛ ≤0.7 , −0.8 ≤αΣ−Ξ−≤0.8(6.63)One can reduce these ranges to some extent by making some plausible assumptionsas discussed in ref.[193]. The main conclusion of that paper is that to test the chiralpredictions of the asymmetry parameters (6.63) a measurement is called for.

In the heavymass formalism, Jenkins et al. [193] have performed a calculation of the imaginary partsof Aab and Bab.

They argue that with the exception of the asymmetry parameter forΣ+ →pγ, good agreement between theory and the available data is obtained.6.6. Nuclear forces and exchange currents79

In this section, I will discuss the recent work of Weinberg and collaborators[194,195,196] on the nature of the nuclear forces and the extension of this to meson–exchange currents by Rho et al. [197,198].On the semi–phenomenological level, nuclear forces are well understood in terms ofmeson exchange.

There is, of course, the long–range pion component first introduced byYukawa [199]. The intermediate range–attraction between two nucleons can be under-stood in terms of correlated two–pion exchange, sometimes parametrized in terms of afictitious scalar–isoscalar σ–meson with mass Mσ ≈550 MeV.

ω–meson exchange givesrise to part of the short–range repulsion and the ρ features prominently in the isovector–tensor channel, where it cuts down most of the pion tensor potential. Weinberg [194,195]has recently discussed the constraints from the spontaneously broken chiral symmetryon the nature of the nuclear forces.

For that, consider the Lagrangian (6.9) includingalso the four–nucleon terms L ¯ΨΨ ¯ΨΨ. Heavy particles are integrated out, their contri-bution is hidden in the values of the pertinent low–energy constants.

Also, consideronly nucleons with momenta smaller than some scale Q. This induces a Q–dependenceof the various coupling constants.

Alternatively, one could work in the framework ofdimensional regularization discussed so far, but the ”Q–language” is more familiar tonuclear physicists. Clearly, one must set Q ≪Mρ ≃m.

Because of this restriction inthe nucleon momenta, it is advantageous to use old–fashioned time–ordered perturba-tion theory in which nucleon propagators are substituted by energy denominators andthe integrations run over the three–momenta. The lowest order Lagrangian mandatedby chiral symmetry is (for the power counting, see below)L =Lππ + LπN + L ¯NN= −12D2 (∂µ⃗π)2 −M 2π2D ⃗π2−¯ΨNi∂0 −m −gADFπ⃗τ · (⃗σ · ⃗∇)⃗π −12DF 2π⃗τ · (⃗π × ∂0)⃗π)ΨN−12CS(¯ΨNΨN)(¯ΨNΨN) + 12CT (¯ΨN⃗σΨN) · (¯ΨN⃗σΨN) + .

. .

(6.64)with ⃗t the isospin generators and ΨN denotes the nucleon field. We have used stere-ographic coordinates on S3 ∼SO(4)/SO(3) ∼SU(2) × SU(2)/SU(2) with D =1 + ⃗π 2/4F 2π.

CS and CT are unknown constants and the ellipsis stands for terms withmore derivatives, pion mass insertions or nucleon fields. Consider the scattering of N,N ≥2, nucleons.

There arises a complication in the power counting due to the existenceof shallow nuclear bound states. One should not consider the S–matrix but rather theeffective potential which is defined via the sum of the connected diagrams without Nnucleon intermediate states.

This stems from the fact that the energy denominatorsof states involving solely N nucleons are small (∼Q2/2m) and these cause the pertur-bation series to diverge at low energies. For a typical pion exchange ladder diagram,this infrared singularity is discussed in detail in ref.[195].

If, however, at least one pion80

exists in the intermediate state, no such problem arises (for a more general discussionof the N nucleon case, see Weinberg [195]). So we are seeking an expansion in the smallparameters Q/m and Mπ/m.

To avoid large factors from time derivatives giving thenucleon mass term, one performs a field redefinition which allows to express the timederivative in terms of other chirally invariant couplings with redefined coefficients,i∂0 −12DF 2π⃗t · (⃗π × ∂0⃗π)ΨN =m + gADFπ⃗t × (⃗σ · ⃗∇)⃗π + . .

.ΨN(6.65)This is similar to the heavy mass formalism of ref.[147]. The power counting goes asfollows.

Consider a vertex of type i with di derivatives or pion mass insertions involvingni nucleon fields. A graph with Vi vertices of type i and L loops scales with ν powersof Q or Mπν = 2 −N + 2L +XiVi(di + ni2 −2)(6.66)which follows after making use of some topological identities.

For any interaction whichis allowed by chiral symmetry (remember that the pion mass counts as order Q), thecoefficient of Vi is non–negative. The lowest order diagrams are tree graphs (L = 0) withvertices which have di = 2 and ni = 0 or di = 1 and ni = 2 or di = 0 and ni = 4.

Thisjust leads to the effective Lagrangian (6.64). One can derive an interaction Hamiltonian(for details, see refs.

[194,195])Hint = Hππint + HπNint + HNNintHNNint = 12CS(¯ΨNΨN)(¯ΨNΨN) + 12CT (¯ΨN⃗σΨN) · (¯ΨN⃗σΨN)(6.67)From this Hamiltonian, one can easily derive the lowest–order static potential betweentwo nucleons,V12(⃗r1−⃗r2) = [CS+CT⃗σ1·⃗σ2]δ(⃗r1−⃗r2)− gAFπ2(⃗t1·⃗t2)(⃗σ1·⃗∇1)(⃗σ2·⃗∇2) Y (|⃗r1−⃗r2|)−(1′ ↔2′)(6.68)with Y (r) = exp(−Mπr)/4πr the standard Yukawa function. This potential containsthe long range one–pion exchange, the pion pair exchange leading to the intermediaterange attraction is hidden in the constant CS.

In fact, to accomodate the small nuclearbinding energies, one has to choose CS large and CT small. This is discussed in detailin ref.[195].

The lowest order approximation of the chirally constrained two–nucleon po-tential can not explain the mysterious intermediate–range attraction and also shows nosign of the repulsive hard core. Ordonez and van Kolck [196] have considered all termswhich are suppressed up to and including (Q/m)2 with respect to the leading potential(6.64).

These include one–loop diagrams and local contact terms with coefficients ofrelative order (Q/m) and (Q/m)2. Again, intermediate states with nucleons only areexcluded.

The low–energy constants accompanying the new contact terms have not yet81

been determined, so that the quantitative value of the approach can not be assessedat present. However, some interesting qualitative features emerge (for explicit expres-sions of the higher order corrections to the two–nucleon potential, see ref.[196]).

In thisframework, the uncorrelated two–pion exchange together with some of the higher ordercontact terms furnishes the intermediate–range attraction, the conventional correlatedtwo–pion exchange (pion rescattering, intermediate ∆states) only appears at next–to–next–to–leading order, (Q/m)4 (compare the discussion in ref.[96]). Furthermore, thepotential is energy–dependent and contains all terms known from phenomenological ap-proaches.

The connection to the meson–exchange potentials can be made if one expressesthe low–energy constants in terms of meson couplings and form factors. Weinberg [194]has also made a very interesting observation concerning the three–nucleon forces.

Tolowest order, these fall into two classes, the one which has only two–body interactionsand the other one with genuine three–body forces. The latter class involves diagramswith exclusively non–linear pion interactions, i.e.

the emission of two pions at one vertex(∼Nππ). Summing up all diagrams of this type, the net result is zero.

Thus, to leadingorder, the three–nucleon potential is entirely build from two–body forces without addi-tional genuine three–body ones. Such a behaviour has indeed been observed in exactcalculations of the three nucleon system in the continuum (deuteron break–up just abovethreshold [200]).

In this reaction, the momenta involved are small and all observableslike the analyzing power etc. can nicely be explained by using only realistic two–bodypotentials and including some charge symmetry breaking.

No genuine three–body forcesseem to be needed. Beyond leading order, there is a three–nucleon potential ∼(Q/m)3[196].

It involves some higher order pion–nucleon interactions and also some contactoperators consisting of six nucleon fields. The quantitative effects of this potential havenot yet been worked out.Meson exchange currents arise naturally in the meson–exchange picture of the nu-clear forces.

An external electromagnetic or axial probe does not only couple to thenucleons (impulse approximation) but also to the mesons in flight. The existence ofthese two–body operators has been verified experimentally [201].

More than 10 yearsago, the so–called ”chiral filter hypothesis” was introduced [202]. It states that theresponse of a nucleus to a long–wavelength electroweak probe is given solely by thesoft–pion exchange terms dictated by chiral symmetry.

Stated differently, all the heav-ier mesons and nucleon excitations are not seen, even up to energies of the order of 1GeV. Why this holds true at such energies has not yet been explained.

Rho [197] hasgiven a simple argument how the ”chiral filter” can occur in nuclei. His lowest orderanalysis follows closely the one of Weinberg [194].

Any matrix–element ME of the ef-fective potential V or of a current Jµ has the form ME ∼Qν F(Q/m), as discussedbefore. In the presence of a slowly varying external electromagnetic field Aµ (or a weakone), the Hamiltonian takes the formHeff= Hππ + HπN + HNN + HextHext =eD2(⃗π × ∂µ⃗π)3 + igA2Fπ¯ΨNγ5γµ(⃗τ × ⃗π)3ΨNAµ + .

. .

(6.69)82

and this additional term Hext modifies the power counting.Since one derivative isreplaced by the external current, the tree graphs (L = 0) with the lowest power ν mustfulfilldi + 12ni + 2 = −1(6.70)which leads to di = 0 and ni = 2 or di = 1 and ni = 0.In contrast to the caseof the nuclear forces, to leading order no four–nucleon contact term contributes. Thismeans that there is no short–ranged two–body current (to lowest order), the exchangecurrent is entirely given in terms of the soft–pion component derived from (6.70).

Thisjustifies the chiral filter hypothesis at tree level. Park et al.

[198] have also investigatedone–loop corrections to the axial–charge operator, the first correction to the pertinentsoft–pion matrix–element is suppressed by (Q/m)2 [195]. The authors of ref.

[198] use theheavy mass formalism which simplifies the calculation considerably. It is shown that theloop corrections are small for distances r ≥0.6 fm, which means that the lowest orderargument of Rho [197] is robust to one–loop order.

A vital ingredient in the calculationis that zero–range interactions between the nucleons are omitted, such interactions aresuppressed in nuclei by the short–range correlations. This is a very nice result.

However,what remains mysterious is why the chiral filter hypothesis works up to so high energies– the answer to this lies certainly outside the realm of baryon CHPT.Finally, Weinberg [203] has recently used these methods to calculate pion–nucleusscattering lengths. In particular, the chiral symmetry result for the πd scattering lengthdoes agree with the more phenomenologically derived ones previously [154] and is thusin agreement with the data.

Here, the strength of the method lies in the fact that theconsistent power counting employed puts the phenomenological result on firm groundsand justifies the assumptions going into its derivation.VII. STRONGLY COUPLED HIGGS SECTOR7.1.

Basic ideasThe standard model of the strong and electroweak interactions is a spectacularlysuccessful theory. However, we do not understand the dynamics underlying the elec-troweak symmetry breaking of SU(2)L × U(1)Y →U(1)em.What we know is thatthis mechanism provides the W and Z bosons with their mass and therefore with theirlongitudinal degrees of freedom.

This indicates that the interactions of longitudinallypolarized gauge bosons will provide us with information about the mechanism triggeringthe symmetry breaking. In the minimal standard model (MSM), which has been stud-ied extensively (see e.g.

the monograph by Gunion et al. [204]), the symmetry breakingproceeds as follows.

The LagrangianLMSM = Lg −V(7.1)83

contains a part which includes Yang–Mills and gauge couplings, gauge–fixing, ghost andmass terms as well as Yukawa couplings, Lg, plus the Higgs potential V . The latter canbe written in an O(4) notation asV (⃗w, H) = λ4(⃗w2 + H2) −v22(7.2)where ⃗w = (w+, w−, z) is a triplet of scalars and H is a fourth scalar field.

In case ofspontaneous symmetry breaking, the vacuum is asymmetric under the O(4), < ⃗w >= 0and < H >= v. Shifting the Higgs field so that it has no vacuum expectation value(vev), H →H −v, one can rewrite (7.2) asV (⃗w, H) = λ4 (⃗w2 + H2)2 + λvH(⃗w2 + H2) + λv2H2(7.3)which shows that there are three massless scalars M⃗w = 0 and the Higgs field hasthe mass M 2H = 2λv2. This is the expected pattern since the O(4) is broken downto O(3), i.e.

there should be (12-6)/2=3 Goldstone bosons. These are then eaten upby the originally massless W ± and Z vector bosons, providing them with their massand longitudinal degrees of freedom since a massless vector field has only two transversecomponents.

Therefore, the dynamics of the longitudinal components VL (from here on, Icollectively denote the W ′s and Z′s by V ) is intimately linked to the symmetry breakingsector. The pattern described above is the MSM Higgs mechanism.

The conventionallyused complex scalar doublet Φ is given by ΦT = (−iw+, (H + v −iz)/√2). From muondecay we know the scale,< H >= v = (√2GF )−1/2 = 246 GeV(7.4)with GF = 1.166·10−5 GeV−2 the Fermi constant.

However, the scalar coupling strengthλ is undetermined and thus the Higgs mass is a free parameter of the model. If the Higgsboson is light, of the order of 100 GeV, the system including also the gauge and mattersector is amenable to perturbative calculations.

This is one possibility. If the Higgs isheavy, say MH ∼1 TeV, the couplings become large and the symmetry breaking sector isstrongly coupled [205,206].

Naive perturbation theory fails and one has to invoke non–perturbative methods. Alternatively, there might be no Higgs boson whatsoever butinstead the symmetry breaking is induced by some strongly coupled theory.

An examplefor this scenario would be the celebrated technicolor [207] which is a scaled up version ofQCD with the replacements Fπ →v and NC →NT C (the numbers of techicolors mightnot be three). Such theories generally have a rich spectrum of low–lying resonances, inthe case of technicolor these are the techni–rho and the pseudo–Goldstone bosons (theones which do not give mass to the V ′s).

For orientation, the techni–rho is expected tohave a mass of mρ,T C =p3/NT C(v/Fπ)mρ = 2.04 TeV for an SU(3)T C theory withtwo technifermion families (for a general discussion, see ref. [208]).84

In the absence of an understanding of the symmetry breaking sector, one has toresort to symmetry arguments if one indeed deals with a strongly coupled system. Thisis what I will assume from here on.

In that case, the EFT methods become useful. Ongeneral grounds, if the underlying dynamics produces resonances, these must couple tothe longitudinal vector bosons.

Depending on the actual masses and widths of theseresonances, they might be seen at the SSC or the LHC. However, it might also happenthat they are too heavy to be detected at the hadron colliders.

In that case, the analogyto QCD is again useful.Although the ρ–meson sits at 770 MeV, it leaves its tracein the particular values of some low–energy constants which are tested at much lowerenergy, say in threshold ππ scattering, √s = 2Mπ ≈Mρ/3. Similarly, one expects anenhancement in VLVL pairs [209] as a signal of the strongly interacting Higgs sector.

Oneparametrizes the symmetry breaking sector by an effective Lagrangian and replaces theHiggs field by an infinite tower of non–renormalizable operators with a priori unknowncoefficients [210]. A particular model of the electroweak symmetry breaking will leadto a definite pattern of the coupling constants.Clearly, at sufficiently low energies,√s ∼1 TeV, only a finite number of terms will contribute since they can be organizedin an energy expansion (as it is the case in QCD).

For doing that, one has to set scales.In analogy to QCD, one expects the scale of symmetry breaking to be Λ ≃4πv ≃3TeV and furthermore estimates the effective theory in the one–loop approximation tobe useful up to energies of approximately √s ≃Λ/2 ≃1.5 TeV. If there are lower–lyingresonances with mass MR < Λ, this window is reduced to √s < MR.

However, this canonly be considered as a very rough estimate.Since in QCD the purest reaction to test the symmetry breaking is ππ scattering,in the strongly coupled Higgs sector much work has been focused on the equivalentreaction VLVL →VLVL. At high energies, one can make use of the so–called equivalencetheorem [211,212].

It relates the S–matrix containing the unphysical Goldstone bosons(w+, w−, z) to the S–matrix containing the physical longitudinal vector bosons. Anymatrix–element M can be decomposed asM(VL, .

. .) = M(⃗w, .

. .) + O(MV /√s)(7.5)where the ellipsis stands for matter fields, transverse gauge bosons and so on.Thetrue S–matrix element containing the longitudinal degrees of freedom VL and any otherphysical particles can be written in terms of the Goldstone bosons w as if these werereal particles.Clearly, the equivalence theorem becomes operative at high energies,√s >> MV .

It is important to notice that it holds to all orders in the scalar coupling λ[212] which is essential to make it useful for a strongly coupled theory. In fact, there aresome subtleties related to eq.(7.5).

Let me just discuss one of them. While the l.h.s.ofeq.

(7.5) is manifestly gauge–independent, the r.h.s. is not.

One can, however, show thatall gauge–dependent terms are multplied by a factor MV /√s (at least) and are thussuppressed. Further subtleties of the equivalence theorem are discussed in ref.

[213].There is one more essential phenomenological ingredient which has to be discussedbefore writing down the effective Lagrangian at next–to–leading order.The MSM85

requires only a breaking from G = SU(2)L × U(1)Y down to H = U(1)em.How-ever, the Higgs potential (7.3) shows all features of a Goldstone realized chiral SU(2),G = SU(2)×SU(2) →H = SU(2), since O(4) ∼SU(2)×SU(2) and O(3) ∼SU(2). Tosee that in more detail, define the field H as a scalar and the ⃗w as pseudoscalars.

Then,V (⃗w, H) is invariant under parity. One can eliminate the scalars via a field redifinition,which is nothing but choosing the complex field Φ in the unitary gauge.

The resultingeffective theory is a non–linear σ–model with F = v. Such a chiral SU(2) is a feature ofa large class of experimentally viable models of the electroweak symmetry breaking. Ithas the property that the ρ–parameter, which measures the ratio of neutral to chargedcurrent amplitudes [214]ρ =M 2WM 2Z cos2 θW= 1 + O(α)(7.6)is protected against strong interaction corrections [215] and only subject to small radia-tive corrections as denoted by the terms of order α (here, α is a genuine symbol for theelectroweak gauge couplings)∗.

In nature, ρ = 0.998 ± 0.0089 [216], i.e. ρ = 1 is fulfilledwithin a few per cent.

In the MSM with G = SU(2)L × U(1)Y , eq. (7.6) holds at treelevel.

However, in that case there is no relation between the third generator proportionalto z and the first two proportional to w± and thus loop corrections will lead to sizeabledeviations from unity. In case of the larger symmetry group G = SU(2)L ×SU(2)C, thethree Goldstone bosons belong to a triplet and thus (7.6) is protected.

This is completelyanalogous to the case of isospin symmetry in two–flavor QCD which protects the vectorcharges against strong renormalization [217]. The additional SU(2)C is called the ”cus-todial” symmetry or one speaks of weak isospin.

In most of the following discussion, Iwill assume such a custodial symmetry or a larger group in which it is embedded. So weare now in the position to write down the pertinent effective Lagrangian and calculatevarious processes.7.2.

Effective Lagrangian at next–to–leading orderIn this section, I will discuss the effective Lagrangian to order p4 assuming theexistence of a custodial SU(2) symmetry. The embedding into a larger group G willalso be discussed briefly.

In contrast to the case of QCD, one has to quantize the externalsources since they are part of a gauge theory. I will work in the Landau gauge (ξ →∞)since in that case the pseudoscalar Goldstone bosons remain massless, Mw = MV /ξ,and they also decouple from the ghost sector.

Furthermore, since ρ is not exactly one innature, one has to allow for some weak isospin breaking. This can come from the quarkmass differences in the doublets (here, only the difference mt −mb or an eventual fourth∗Clearly, one has to define ρ differently if one chooses eq.

(7.6) to define the weak mixingangle.86

generation is relevant), hypercharge gauge boson loops and so on. To lowest order p2,the effective Lagrangian consists of two terms [210,218,219],L(2) = v24 Tr (DµU †DµU) + ∆ρv28 Tr [τ 3(U †DµU)]2(7.7)where U = exp(iτ iwi/v) collects the Goldstone bosons and the covariant derivativereadsDµU = ∂µU + i2gWµU −i2g′BµUτ 3(7.8)with Wµ = W iµτ i.

The SU(2)L and U(1)Y gauge couplings are denoted by g and g′,respectively. In the unitary gauge U = 1, the first term of (7.7) reduces to the standardbilinear gauge boson termsL(2)(U = 1, ∆ρ = 0) = v24g22 W iµW iµ + g′22 BµBµ −gg′W 3µBµ(7.9)so that MW = gv/2 leads to v = 246 GeV as stated before.

Diagonalization of (7.9) givesthe massless photon and the massive Z. The second term in (7.7) brings the expectedshift of the Z mass so that ρ ̸= 1.

In fact, to leading order in the quark masses, onefinds ∆ρ ∼(mt −mb)2/v2 ∼m2t/v2 [220]. Ramirez [221] has calculated ∆ρ in twoparticular models, one with a heavy scalar and the other with a techni–rho.

While theeffects are of opposite sign in the two models, the contribution to ∆ρ is too small tobe detected at present. However, from this discussion it becomes clear that one has tocleanly separate the known physics (here the m2t contribution) from the unknown onerelated to the mechanism of the symmetry breaking.At next–to–leading order, there are much more terms.

These have been enumeratedby Longhitano [218] some ten years ago. Neglecting custodial symmetry breaking, theorder p4 Lagrangian readsL(4) =L116π2Tr (DµU †DµU)2 +L216π2 Tr (DµU †DνU) Tr (DµU †DνU)−igL9L16π2 Tr (W µνDµUDνU †) −ig′L9R16π2 Tr (BµνDµU †DνU)+ igg′L1016π2Tr (UBµνU †Wµν)(7.10)with 2Bµν = (∂µBν −∂νBµ)τ 3 and 2W µν = ∂µWν −∂νWµ −ig[Wµ, Wν]/2.

We haveused the convention of Falk et al. [222] and pulled out a factor (1/16π2) so that the Liare numbers of order one.

In vectorial theories like e.g. SU(2)L×SU(2)C →SU(2)L+C,one has L9R = L9L.

Clearly, the terms proportional to L1,2 are related to Goldstoneboson scattering while the last three involve external electroweak gauge bosons. There87

are also O(p4) symmetry breaking terms which can be found in ref.[218]. They will notbe discussed in what follows.Let me briefly consider theories with larger symmetry groups [223] with an em-bedding of the custodial symmetry.

In that case, there are additional massive scalars,the so–called pseudo–Goldstone bosons. In most models, these are relatively light andtypically carry color.

So one has to enlarge the covariant derivative (7.8) to include theembedding of SU(3)c × SU(2)L × U(1)Y into the larger group G. Let SU(3), SU(2)and U(1) be generated by Xα = XαAT A, Xi = XiAT A and X = XAT A, respectively.For G = G × G, the pertinent covariant derivative readsDµU = ∂µU + i2gW iµXiU −i2g′BµUX + i2gsGαµ[Xα, U](7.11)with U ∈G, Gαµ the SU(3) color gauge field and gs the strong coupling constant. Onealso has new operators in the effective Lagrangian.

First, there is a mass term for thepseudo–Goldstone bosons, L(2)M = −Tr (UMU †M †), with M the mass matrix of thepseudo–Goldstone bosons.At order p4, one has the following new operators for anSU(2N) × SU(2N) global symmetry (N > 1) [224]L(4)new =L316π2 Tr (DµUDµU †DνUDνU †) + L416π2 Tr (DµUDνU †DµUDνU †)−igsL1316π2 Gαµν Tr [Xα(DµUDνU † + DµU †DνU)](7.12)This extension becomes important in the context of VL pair production via gluon fu-sion. Notice that Soldate and Sundrum [233] have argued that in such models the scaleof symmetry breaking is not 4πv, but rather 4πv/N.

Let us now apply the effectiveLagrangian to various physical processes.7.3. Longitudinal vector boson scatteringIn this section, I will be concerned with elastic scattering of longitudinally polarizedvector bosons which is the equivalent to ππ scattering in QCD.

After some generalremarks on the scattering amplitude, I will discuss some physics issues like e.g. howthe longitudinally polarized gauge bosons can be produced at hadron colliders like theSSC or the LHC.

I will keep the discussion short but provide sufficient references for thereader who wants to go deeper into this subject.At low energies, MV << √s << 4πv (or MR, where MR denotes the mass ofthe lowest resonance), one can derive a low–energy theorem for the scattering processVLVL →VLVL in complete analogy with Weinberg’s prediction for ππ scattering [62]. Itis favorable to work in the Landau gauge so that the Goldstone bosons remain massless.Furthermore, one can make use of the equivalence theorem.

Therefore, I will frequentlyinterchange the symbols ′V ′L and ′w±, z′. To lowest order, the scattering amplitude can88

be given uniquely in terms of a single function A(s, t, u) of the pertinent Mandelstamvariables (with s + t + u = 0),M(w+w−→zz) = A(s, t, u) = sv2M(w+w−→w+w−) = −uv2 , M(zz →zz) = 0(7.13)which agress with eq. (4.3) for Fπ →v and holds for ρ = 1.

The other channels, i.e.elastic w±z, w+w+ and w−w−scattering, follow by crossing. If there is no custodialsymmetry or if it is broken (ρ ̸= 1), Chanowitz et al.

[225] have generalized (7.13),M(w+w−→zz) = sv21ρ , M(w+w−→w+w−) = −uv24 −3ρ(7.14)and the third prediction in (7.13) remains unaffected. The empirical information ρ ≈1tells us that in the low energy region, symmetry requirements uniquely determine theVLVL scattering amplitudes in terms of a single scale, v. It is instructive to see how inthe MSM this LET emerges.

Exchange of a scalar Higgs field givesA(s, t, u) = −M 2Hv2ss −M 2H= sv2 +s2v2M 2H+ O(s3)(7.15)The first term in the expansion for s << v2 agrees with the LET, the second one dependson the Higgs mass, which in this case is the information about the symmetry breakingsector.Therefore, the idea is to measure deviations from the low–energy behaviour∼s/v2 and try to relate these to the models of the dynamics underlying the symmetrybreaking.For the order p4 chiral Lagrangian (7.7,7.10) (with ∆ρ = 0), the VLVL scatteringamplitude contains two unknown coefficients related to the low–energy constants L1 andL2. This is, of course, completely analogous to Lehmann’s [65] one–loop analysis of ππscattering in the chiral limit.

At next–to–leading order, the scattering amplitude isA(s, t, u) = sv2 +116π2v48L1(µ)s2 + 4L2(µ)t2−s22 ln−sµ2−112(3t2 + u2 −s2) ln−tµ2+ (t ↔u)(7.16)Notice that there might be additional constant terms in the p4 pieces depending onthe renormalization scheme one uses (see ref. [219] for a detailed discussion).

It is nowinstructive to look at the various partial waves for given isospin (I) and angular momen-tum (J). In the I = 0 S–wave, the contribution 11L1(µ) + 7L2(µ) enters whereas theI = J = 1 amplitude is sensitive to L2(µ)−2L1(µ) (which actually is a scale–independent89

quantity) and the exotic S–wave measures L1(µ) −2L2(µ) [226]. The unitarity effectsare entirely determined by the scale v and the scale of dimensional regularization, µ.In the MSM, the one loop effects have been calculated in refs.[227].

Apart from thetree level Higgs exchange (7.15) one has corrections of the type ln(M 2H). To get an ideain the case of strong coupling, let us imagine models which contain nearby resonancesthat saturate the low–energy constants (as already discussed in section 2).

For a heavyscalar–isoscalar Higgs boson, one finds after matching at the scale MH [13,224]L1(µ) = 64π33ΓHv4M 5H+ 124 logM 2Hµ2L2(µ) = 112 logM 2Hµ2(7.17)where ΓH = 3M 3H/32πv2 is the standard model width for the Higgs. For a Higgs withmass MH = 2 TeV, one finds at the scale µ = 1.5 TeV, L1(µ) = 0.33 and L2(µ) = 0.01(these are the renormalized values).

For a model with an isovector–vector exchange, onefinds at the scale Mρ [13,224]L1(µ) = −192π3 Γρv4M 5ρ+ 124 logM 2ρµ2L2(µ) = −L1(µ) + 18 logM 2ρµ2(7.18)with Γρ the pertinent width. Scaling up the QCD ρ–meson as described before, onefinds L1(µ) = −0.31 and L2(µ) = 0.38 at µ = 1.5 TeV.

However, there are othercontributions to the Li.The µ–dependence is actually generated by the Goldstoneboson loops and then there is also a fermionic contribution, which can be calculated.For example, Dawson and Valencia [228] have considered the effect of a heavy top quark(mt ≤200 MeV) and found it contributes insignificantly. This again underlines the factthat one has to subtract the known physics from the inferred low–energy constants toget a handle on the underlying dynamics.

Let us consider the effect of the particularvalues of the L1,2 as given by eqs. (7.17,7.18) in VLVL scattering [226,229,230,231].

Inthe case of a heavy Higgs, the I = J = 0 partial wave is enhanced and T 11 is suppressed.The effect is opposite in the case of an isovector–vector exchange, one finds suppressionin T 00 and enhancement in T 11 . Notice also that the process zz →zz is very sensitiveto the actual values of L1,2 since its tree level amplitude vanishes.

The Higgs modelwould predict a positive amplitude, while technicolor keeps it essentially zero since thetechni–rho does not couple to neutral gauge bosons. For more detailed information onthese topics, including also a discussion of various unitarization models, I refer to therecent review by Hikasa [232].The question arises how to access V V scattering at hadron colliders?In mostcalculations, one assumes factorization of the production of the VLVL pairs, their scat-tering and their subsequent decay.Let us focus on the first part of this chain.In90

Fig. 15:Mechanism of producing longitudinal vector gauge boson pairs.

Theseare a) quark–antiquark annihilation, b) gluon fusion and c) vector boson fusion.Solid lines denote (anti)quarks, wiggly lines gluons and dashed lines vectorbosons.pp colliders, there are essentially three mechanisms to produce vector boson pairs, seefig.15. The annihilation of light quarks and antiquarks (fig.15a) gives rise to mostlytransversely polarized V ′s and it is therefore an important background.

Appropriatecuts have to be chosen to separate it from the longitudinal pairs. However, as it is thecase of the ρ–resonance in e+e−annihilation, the produced longitudinal pairs are mostlyin an isospin–one state [230] making this process sensitive to models with techni–rho’sor alike.

Also, if there are more general couplings with L9L ̸= L9R, this mechanismcan be of importance [222]. Second, there is the fusion of two gluons (fig.15b).

Onlyif the gluons couple to a loop of heavy quarks, a sizeable amount of VLVL pairs is pro-duced. This mechanism is sensitive to isospin–zero resonances and thus a prime wayof exploiting a heavy Higgs boson.

Also, in models with additional pseudo–Goldstonebosons, it is considerably enhanced due to large color factors [223]. Third, there is theso–called vector boson fusion as shown in fig.15c.

It is, of course, only relevant if theHiggs sector is strongly interacting [212]. This mechanism gives the cleanest signal if oneis able to isolate it.

The luminosities of the initial V V pair (transverse or longitudinal)are generally calculated in the ”effective W–approximation” [234], in which the V ′s are91

considered as partons, i.e. as on–shell, physical bosons.

Since they are radiated offthe(anti)quarks, one also needs the quark distribution functions in the incoming hadrons.For a detailed study of this topic, see e.g. ref.

[208].After the scattering, one has to detect the V ′s and separate the scattering processfrom the background. The most detailed study of the reaction pp →VLVLX has beenperformed by Bagger, Dawson and Valencia [223,224,235] and Bagger [236] has givenrate estimates for the LHC and the SSC and discussed the various cuts, tags and vetoeswhich go into the detection of the ”gold–plated” decays W ± →ℓ±¯νℓand Z →ℓ+ℓ−(for ℓ= e, µ) in each of the final states W +W −, W +Z, W +W + and ZZ.

The taggingand veto methods are discussed in detail in refs.[237,238]. One finds that the events areclean, but have a low rate if there are only high mass resonances, say MR ≥2 TeV.However, even in the case of a standard model Higgs with MH = 1 TeV, the eventrates are not large.

Isospin–zero resonances are most cleanly seen in the W +W −andZZ channels, while the W +Z channel would be dominated by isospin–one resonances.Models without low mass resonances tend to be most visible in the W +W + final state.This means that there is some complementarity in the pertinent signals differentiatingthe various models of the strongly coupled Higgs sector. For typical cuts and the nominalSSC and LHC luminosities and energies (these are 104 and 105 pb−1 and √s = 40 and16 TeV for the SSC and the LHC, respectively) and assuming mt = 140 GeV, theevent rates are of the order of O(10) per year and somewhat larger than the expectedbackground for most models.

Clearly, only high statistics experiments will be able tounravel the nature of the mechanism triggering the electroweak symmetry breaking inthe case of strong coupling [236]. It is also instructive to see to which kind of newphysics the various final states are sensitive in terms of the low–energy constants.

Ifthe VLVL pairs are created via ¯qq annihilation, the W +L W −L and W ±L ZL final states aresenstive to L9R,L because the anomalous three gauge boson vertex enters here. Thiswas first pointed out by Falk, Luke and Simmons [222] for hadron colliders like the SSCor the LHC.

They estimated from the phenomenology of WZ production that the SSCwill be sensitive to −16 ≤L9L ≤7 and the LHC sets the limits −22 ≤L9L ≤12 fromthe transverse momentum distribution of the Z′s produced in pp →W ±Z + X. Thechannel W ±γ gives limits on the sum ˆL = L10 + (L9L + L9R)/2, but these are only ofthe order |ˆL| ≤50 (60) for the SSC (LHC). A similar study was recently performed byDobado and Urdiales [239].

The trilinear gauge boson vertex was also considered byHoldom [240] who investigated technicolor effects on the reaction e+e−→W +W −. Inthe case of vector boson fusion, one is naturally sensitive to the couplings L1,2.

Gluonfusion plays a role in the case of colored pseudo–Goldstone bosons as already stated andwould enhance the W +L W −L and ZLZL final states considerably. Further discussions ofthese topics, concerning also e+e−machines, can be found in refs.[232,241].

Finally,constraints on the symmetry breaking sector from a Adler–Weisberger type sum rulehave been discussed by Weinberg [242] and Pham [243].7.4. Electroweak radiative corrections92

In this section, I will be concerned with the use of EFT methods for calculatingelectroweak radiative corrections. The high–precision measurements of weak–interactionparameters at LEP have spurred much interest in precise calculations of these radiativecorrections from both within and beyond the standard model (SM).

For the SM, thesehave already been calculated using conventional techniques [209,244]. For an introduc-tion on these topics, I refer to the lectures by Peskin [245].

In particular, much workhas been focused on the so–called ”oblique” corrections, that is corrections to the gaugeboson propagators (vacuum polarization effects) [246]. There are several reasons why itis advantagous to consider oblique corrections.

First, if there are heavy particles whichdo not directly couple to the light fermions, they can only be detected via their loopcontributions to the gauge boson propagators (for the presently operating accelerators).This is due to the fact that in most models of the electroweak symmetry breaking thedecoupling theorem [247] is expected to be inoperative. Second, vacuum polarizationeffects are universal.

They do not depend on the specific process under consideration incontrast to e.g. vertex or box corrections.

Furthermore, for reactions involving exclu-sively light external fermions, one can neglect their masses as compared to the Z massand therefore only has to take into account the transverse (gµν) part of the gauge bosonpropagator. Following Peskin and Takeuchi [248,249], the oblique corrections can beparametrized in terms of three combinations of the gauge boson self–energies and theirderivatives,αS = 4e2[Π′33(0) −Π′3Q(0)]αT =e2s2c2M 2Z[Π11(0) −Π33(0)]αU = 4e2[Π′11(0) −Π′33(0)](7.19)with s = sin θW , c = cos θW and the indices ’1,2,3,Q’ refer to the W, Z and γ particles.The quantity S is isospin–symmetric, whereas T (which is nothing but ∆ρ/α alreadydiscussed) and U are isospin–asymmetric.

Since U is related to isospin–breaking in thederivatives of the self–energies, it is supposed to be small in models with a custodialsymmetry and often neglected.Notice that Peskin and Takeuchi [249] define theseparameters as deviations from the SM ones for a given reference point mt = 150 GeVand MH = 1 TeV. The relation to the epsilon parameters of Altarelli et al.

[250] isǫ1 = αT, ǫ2 = −αU/4s2 and ǫ3 = αS/4s2. The recent analysis of these authors givesS = 0.1 ± 1.1, T = −0.06 ± 0.69 and U = 1.4 ± 1.3 using exclusively LEP results.

Ifone furthermore includes low–energy data from atomic parity violation in Cs [251], onefinds S = −0.34 ± 0.68, T = −0.03 ± 0.48 and U = 0.78 ± 0.98.The use of EFT methods in the analysis of oblique corrections was pioneered byGolden and Randall [252], Holdom and Terning [253], Holdom [254] and Dobado etal. [255].The most lucid discussion of these methods can be found in the paper ofGeorgi [256] to which I will return later.

First, let me give an elementary discussion ofthe physics behind the parameter S and its relation to the chiral Lagrangian. In thebasis of the non–physical fields A3µ and Bµ it is most simple to calculate the one–loop93

diagram since this gives exactly the mixing which defines S. The graph is divergent, itcan be made finite by a counterterm of the typeL10gg′BµνW µνaTr [U †τ3Uτa](7.20)from which it immediately follows thatS = −16πLr10(7.21)Obviously, one has to specify the renormalization prescription to make this equationprecise. As already stressed, S contains a part coming from the SM (denoted by SSM)and eventually a part due to physics from some higher scale Λ ≫MZ (denoted bySNP ).

A particular model for the strongly coupled Higgs sector will lead to a certainvalue of L10 and thus of SNP [253,254]. SSM gets contributions from the quarks andalso the SM Higgs.

For MH = 1 TeV, Altarelli et al. [250] give SSM ≃0.65 whereasSint finds SSM ≃−0.21 [219].

It is not yet clear what the source of this discrepancyis. In any case, there is not much room left for non–SM physics if one compares toSexp = −0.34 ± 0.68.

As stressed in refs. [248,254], simple technicolor models give rise toa positive contribution to SNP , approximately [249,257]SNP ≃0.3NT F2NT C3(7.22)for a model with NT C technicolors and NT F technifermions.

Since the empirical valueof S is negative, QCD–like technicolor theories with a large technisector seem to beruled out. However, it is possible to construct models which lead to negative valuesof S [256,258].

In any case, a fully consistent technicolor theory is not yet availableand therefore the emiprical information on S, T and U should be used to constrain anymodel of the electroweak symmetry breaking. Finally, notice that S can be written interms of a Weinberg sum rule [219,248,257] in complete analogy to the QCD case forL10 or ¯ℓ5 [13,14].Let me now return to the paper of Georgi [256].First, it is pointed out whythe EFT formalism is useful in the calculation of radiative corrections.

These reasonsare clarity, simplicity and, most important, generality. The EFT allows one to pickup the relevant operators and avoids most complications due to scheme dependence,which is the case in the conventional approach.

When one integrates out the particlesheavier than some scale Λ, one produces a tower of non–renormalizable operators in theEFT. This integrating out of the heavy particles in general leads to an effective actionwhich is highly non–local.

To arrive at the effective Lagrangian, which is local, onehas to perform an operator product expansion. At the scale Λ, one has to match thephysics in the two theories.

In that way, the low–energy theory keeps the knowledgeabout the heavy particles no longer present as active degress of freedom. This matchingleads to a power dependence on the heavy particle masses.In addition, there is a94

logarithmic dependence on these masses which originates from using the renormalizationgroup to evolve the EFT to some smaller scale µ, µ < Λ.The general problem ofmatching in effective field theories which are not necessarily separated by large scaleshas been discussed by Georgi [259]. The proper matching conditions disentangle theshort–distance from the long–distance physics.

While the former is incorporated in thecoefficients of the effective Lagrangian, the latter remains, of course, explicit in the EFT.In ref. [256], a general analysis of electroweak radiative corrections is performed undera few assumptions (custodial symmetry, no extra Goldstone bosons and no interactionswhich let the light quarks and leptons participate in the mechanism of the symmetrybreaking).

First, the energy domain MH > Λ > mt is considered. The leading termsbeyond the ones from the standard model can be organized in powers of the U(1) gaugecoupling.

At one loop, one has a term like (7.20) contributing to S and an operatorfrom virtual gauge boson exchange contributing to T plus a whole tower of higher order(suppressed) operators. Going down to the regime where mt > Λ > MZ, there appearadditional contributions to S and T, the most important one arises when one integratesout the t–quark.

At this point, it is obligatory to differentiate between the effectiveaction and the effective Lagrangian since the b–quark, which forms a doublet with thet, is retained in the EFT. As an example, a calculation is performed for large αT, sincethis is the only oblique correction that scales with m2t.

It is straightforward to derivethe non–perturbative relationM 2WM 2Z= ρ⋆21 +s1 −4πα⋆√2GF M 2Zρ⋆(7.23)with ρ⋆= 1+αT and α⋆= α(MZ). This relation was originally conjectured in ref.

[260],but it can be derived in a much simpler fashion using the EFT approach.Furthercorrections to eq. (7.23) are also discussed in ref.[256].

In any case, I strongly advice thereader to work through that paper.Finally, let me just point out that the use of EFT methods in the calculation ofradiative electroweak corrections is only in its beginning stage and much more workremains to be done.VIII. MISCELLANEOUS OMISSIONSHere, I essentially assemble references for some of the topics not covered in thepreceeding sections.

However, due to the rapid developments in the field I do not evenattempt to offer a complete list but rather refer the reader to his/her favorite data base.• More mesons : One of the central topics of CHPT was, is and will be the dynamicsand interactions of the pseudoscalar Goldstone bosons at low and moderate energies.Of particular interest are the reactions without counterterm contributions at next–to–leading order since they can serve as a clean test of the chiral sector of QCD.These are K0L →π0γγ [99], KS →γγ [262] and γγ →π0π0 [43] as well as the95

related process leading to the pion electromagnetic polarizabilities [263]. And, ofcourse, there are plentiful pion, kaon and eta semi– and non–leptonic decays whichhave not yet been measured with a high precision.

A thorough discussion of thiscan be found in the user’s guide to the ”chiral temple”, known to the non–chiralworld also as the Frascati Φ–factory DAΦNE [265]. For further references, see alsothe lectures by Ecker [266] and the recent talk by Leutwyler [267].• Union of heavy quark and chiral symmetries : In section 6.2, the nucleon wasdescribed as a very heavy (static) source surrounded by a cloud of light pions.

Thisis the physical picture underlying the so–called ”heavy quark symmetries” of mesonsconsisting of one very heavy (c, b) quark and one very light (u, d, s) antiquark. Inthe limit of the heavy mass becoming infinite, this symmetry becomes exact andone can e.g.

derive relations between certain properties of the pseudoscalar D, Bmesons with their vectorial excitations D⋆, B⋆. For the emission of pions or kaonsfrom these mesons, e.g.

D⋆+ →D0π+ or B →D⋆πlν, one works with an effectiveLagrangian which unites the heavy quark and the chiral symmetry of QCD. This avery new and wide field, and some first work has been reported in refs.

[268].• Large Nf : QCD becomes much simpler if one considers the case of a large numberof colors, NC →∞[26,27,28]. Similarly, one hopes to gain a deeper understand-ing of the chiral expansion by considering the artificial limit of a large number ofmassless (light) flavors.Some pertinent papers are given in ref.[269].

However,after approximately ten years of experience in the context of Skyrme–type models,which are based on a 1/NC expansion, one is inclined to view this developmentrather sceptically. Also, recent lattice studies seem to indicate a qualitatively dif-ferent behaviour for theories with many flavors.• First reading : There exist several introductory lectures concerned with CHPTand its applications.Let me mention here the ones by Donoghue [270], Ecker[266], Gasser [182] and Leutwyler [271].

More specialized lectures are the ones byTruong [272], Jenkins and Manohar [149] and the author [144]. There are also afew books on the market which deal with EFT methods and CHPT.

First, thereis the classical book by Georgi [21] and more recently, a broader exposition hasbeen given by Donoghue, Golowich and Holstein [273]. Finally, a state of the artsummary as of the fall of 1991 is given in the workshop proceedings [274].AcknowledgementsI express my thanks to all my collaborators and everybody who taught me aboutthe material discussed.

I am grateful to V´eronique Bernard for a careful reading of themanuscript. I also wish to thank all members of the Instiute for Theoretical Physics atthe University of Berne for creating most stimulating working conditions.96

APPENDIX A: THE CASE OF SU(2) × SU(2)In the two–flavor case, it is most convenient to work with real O(4) fields of unitlength, e.g. the pions are collected in U A(x) = (U 0(x), U i(x)) with U T U = 1.

Thepertinent covariant derivatives are∇µU 0 = ∂µU 0 + aiµU i∇µU i = ∂µU i + ǫiklvkµU l −aiµU i(A.1)disregarding the isoscalar vector and axial currents. Defining furtherχA = 2 ˜B(s0, pi) , ˜χA = 2 ˜B(p0, −si) ,(A.2)where the value of ˜B is slightly different from the one of B (see ref.[14]).

The next–to–leading order chiral Lagrangian reads:Leff=7Xi=1ℓi Pi +3Xj=1hj ˜Pj(A.3)with (we do not exhibit the high–energy terms)P1 = Tr (∇µU T ∇µU)2P2 = Tr (∇µU T ∇νU) Tr (∇µU T ∇νU)P3 =Tr (χT U)2P4 = Tr (∇µχT ∇µU)P5 = Tr (U T FµνF µνU)P6 = Tr (∇µU T Fµν∇νU)P7 =Tr (˜χT U)2(A.4)One can introduce scale–independent low–energy constants ¯ℓi viaℓri =γi32π2¯ℓi + ln(M 2/µ2)(i = 1, . .

., 6)(A.5)withγ1 = 13 , γ2 = 23 , γ3 = −12 , γ4 = 2 , γ5 = −16 , γ6 = −13 . (A.6)Notice that ℓ7 is not renormalized.

The explicit relation of these low–energy constantsto the Lr1, . .

., Lr10 is spelled out in ref. [14].97

APPENDIX B: SU(3) MESON-BARYON LAGRANGIANHere, I wish to provide the necessary definitions for the three flavor meson–baryonsystem. It is most convenient to write the eight meson and baryon fields in terms ofSU(3) matrices M and B, respectively,M =1√21√2π0 +1√6ηπ+K+π−−1√2π0 +1√6ηK0K−¯K0−2√6η(B.1a)B =1√2Σ0 +1√6ΛΣ+pΣ−−1√2Σ0 +1√6ΛnΞ−Ξ0−2√6Λ(B.1b)withU(M) = u2(M) = exp{2iM/F}(B.2)and the covariant derivative acting on B readsDµB = ∂µB + [Γµ, B]Γµ = 12u†[∂µ −i(vµ + aµ)]u + u[∂µ −i(vµ −aµ)]u†(B.3)with vµ and aµ external vector and axial–vector sources.

Under SU(3)L × SU(3)R, Band DµB transform asB′ = KBK† , (DµB)′ = K(DµB)K†(B.4)It is now straightforward to construct the lowest–order O(p) meson–baryon Lagrangian,L(1)MB = Tri ¯BγµDµB −◦m ¯BB + 12D ¯Bγµ{uµ, B} + 12F ¯Bγµγ5[uµ, B](B.5)where◦m stands for the (average) octet mass in the chiral limit. The trace in (B.5)runs over the flavor indices.

Notice that in contrast to the SU(2) case, one has twopossibilities of coupling the axial uµ to baryon bilinears. These are the conventionalF and D couplings subject to the constraint F + D = gA = 1.26.

At order O(p2)the baryon mass degeneracy is lifted by the terms written in eq.(6.61). However, thereare many other terms at this order.

If one works in the one–loop approximation, onealso needs the terms of order O(p3). The complete local effective Lagrangians L(2)MB andL(3)MB are given by Krause [143].

The extension of this to the heavy mass formalism isstraightforward, I refer to the article by Jenkins and Manohar [149] (which, however,contains not all terms of L(2)MB and none of L(3)MB).98

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